text:
#include <vector>
#include <string>
#include <cmath>
#include "math.h"
#include <Eigen/Dense>
#include <stdlib.h>

#include "PCSAFTBackend.h"
#include "Backends/Helmholtz/VLERoutines.h"

using std::vector;

/*
References
----------
* J. Gross and G. Sadowski, “Perturbed-Chain SAFT:  An Equation of State
Based on a Perturbation Theory for Chain Molecules,” Ind. Eng. Chem.
Res., vol. 40, no. 4, pp. 1244–1260, Feb. 2001.
* M. Kleiner and G. Sadowski, “Modeling of Polar Systems Using PCP-SAFT:
An Approach to Account for Induced-Association Interactions,” J. Phys.
Chem. C, vol. 111, no. 43, pp. 15544–15553, Nov. 2007.
* Gross Joachim and Vrabec Jadran, “An equation‐of‐state contribution
for polar components: Dipolar molecules,” AIChE J., vol. 52, no. 3,
pp. 1194–1204, Feb. 2006.
* A. J. de Villiers, C. E. Schwarz, and A. J. Burger, “Improving
vapour–liquid-equilibria predictions for mixtures with non-associating polar
components using sPC-SAFT extended with two dipolar terms,” Fluid Phase
Equilibria, vol. 305, no. 2, pp. 174–184, Jun. 2011.
* S. H. Huang and M. Radosz, “Equation of state for small, large,
polydisperse, and associating molecules,” Ind. Eng. Chem. Res., vol. 29,
no. 11, pp. 2284–2294, Nov. 1990.
* S. H. Huang and M. Radosz, “Equation of state for small, large,
polydisperse, and associating molecules: extension to fluid mixtures,”
Ind. Eng. Chem. Res., vol. 30, no. 8, pp. 1994–2005, Aug. 1991.
* S. H. Huang and M. Radosz, “Equation of state for small, large,
polydisperse, and associating molecules: extension to fluid mixtures.
[Erratum to document cited in CA115(8):79950j],” Ind. Eng. Chem. Res.,
vol. 32, no. 4, pp. 762–762, Apr. 1993.
* J. Gross and G. Sadowski, “Application of the Perturbed-Chain SAFT
Equation of State to Associating Systems,” Ind. Eng. Chem. Res., vol.
41, no. 22, pp. 5510–5515, Oct. 2002.
* L. F. Cameretti, G. Sadowski, and J. M. Mollerup, “Modeling of Aqueous
Electrolyte Solutions with Perturbed-Chain Statistical Associated Fluid
Theory,” Ind. Eng. Chem. Res., vol. 44, no. 9, pp. 3355–3362, Apr. 2005.
* L. F. Cameretti, G. Sadowski, and J. M. Mollerup, “Modeling of Aqueous
Electrolyte Solutions with Perturbed-Chain Statistical Association Fluid
Theory,” Ind. Eng. Chem. Res., vol. 44, no. 23, pp. 8944–8944, Nov. 2005.
* C. Held, L. F. Cameretti, and G. Sadowski, “Modeling aqueous
electrolyte solutions: Part 1. Fully dissociated electrolytes,” Fluid
Phase Equilibria, vol. 270, no. 1, pp. 87–96, Aug. 2008.
* C. Held, T. Reschke, S. Mohammad, A. Luza, and G. Sadowski, “ePC-SAFT
revised,” Chem. Eng. Res. Des., vol. 92, no. 12, pp. 2884–2897, Dec. 2014.
*/

namespace CoolProp {

PCSAFTBackend::PCSAFTBackend(const std::vector<std::string> &component_names, bool generate_SatL_and_SatV) {
    N = component_names.size();
    components.resize(N);
    ion_term = false;
    polar_term = false;
    assoc_term = false;
    water_present = false;
    for (unsigned int i = 0; i < N; ++i){
        components[i] = PCSAFTLibrary::get_library().get(component_names[i]);
        // Determining which PC-SAFT terms should be used
        if (components[i].getZ() != 0) {
            ion_term = true;
        }
        if (components[i].getDipm() != 0) {
            polar_term = true;
        }
        if (components[i].getVolA() != 0) {
            assoc_term = true;
        }
        if (components[i].getCAS() == "7732-18-5") {
            water_present = true;
            water_idx = i;
        }
    }

    // Set the components and associated flags
    is_pure_or_pseudopure = (N == 1);

    // loading interaction parameters
    std::string kij_string;
    std::string kijT_string;
    if (is_pure_or_pseudopure) {
        this->mole_fractions = std::vector<CoolPropDbl>(1, 1);
    }
    else {
        k_ij.resize(N*N, 0.0);
        k_ijT.resize(N*N, 0.0);
        for (unsigned int i = 0; i < N; ++i){
            for (unsigned int j = 0; j < N; ++j){
                if (i != j) {
                    kij_string = PCSAFTLibrary::get_library().get_binary_interaction_pcsaft(components[i].getCAS(), components[j].getCAS(), "kij");
                    kijT_string = PCSAFTLibrary::get_library().get_binary_interaction_pcsaft(components[i].getCAS(), components[j].getCAS(), "kijT");
                    k_ij[i*N+j] = atof(kij_string.c_str());
                    k_ijT[i*N+j] = atof(kijT_string.c_str());
                }
            }
        }
    }

    if (generate_SatL_and_SatV) {
        bool SatLSatV = false;
        SatL.reset(this->get_copy(SatLSatV));
        SatL->specify_phase(iphase_liquid);
        SatV.reset(this->get_copy(SatLSatV));
        SatV->specify_phase(iphase_gas);
    }

    // Set the phase to default unknown value
    imposed_phase_index = iphase_not_imposed;
    _phase = iphase_unknown;
}

PCSAFTBackend::PCSAFTBackend(const std::vector<PCSAFTFluid> &components_in, bool generate_SatL_and_SatV) {
    components = components_in;
    N = components.size();
    // Determining which PC-SAFT terms should be used
    ion_term = false;
    polar_term = false;
    assoc_term = false;
    water_present = false;
    for (unsigned int i = 0; i < N; ++i){
        if (components[i].getZ() != 0) {
            ion_term = true;
        }
        if (components[i].getDipm() != 0) {
            polar_term = true;
        }
        if (components[i].getVolA() != 0) {
            assoc_term = true;
        }
        if (components[i].getCAS() == "7732-18-5") {
            water_present = true;
            water_idx = i;
        }
    }

    // Set the components and associated flags
    is_pure_or_pseudopure = (N == 1);

    // loading interaction parameters
    std::string kij_string;
    std::string kijT_string;
    if (is_pure_or_pseudopure) {
        this->mole_fractions = std::vector<CoolPropDbl>(1, 1);
    }
    else {
        k_ij.resize(N*N, 0.0);
        k_ijT.resize(N*N, 0.0);
        for (unsigned int i = 0; i < N; ++i){
            for (unsigned int j = 0; j < N; ++j){
                if (i != j) {
                    kij_string = PCSAFTLibrary::get_library().get_binary_interaction_pcsaft(components[i].getCAS(), components[j].getCAS(), "kij");
                    kijT_string = PCSAFTLibrary::get_library().get_binary_interaction_pcsaft(components[i].getCAS(), components[j].getCAS(), "kijT");
                    k_ij[i*N+j] = atof(kij_string.c_str());
                    k_ijT[i*N+j] = atof(kijT_string.c_str());
                }
            }
        }
    }

    if (generate_SatL_and_SatV) {
        bool SatLSatV = false;
        SatL.reset(this->get_copy(SatLSatV));
        SatL->specify_phase(iphase_liquid);
        SatV.reset(this->get_copy(SatLSatV));
        SatV->specify_phase(iphase_gas);
    }

    // Set the phase to default unknown value
    imposed_phase_index = iphase_not_imposed;
    _phase = iphase_unknown;
}

void PCSAFTBackend::set_mole_fractions(const std::vector<CoolPropDbl> &mole_fractions)
{
    if (mole_fractions.size() != N)
    {
        throw ValueError(format("size of mole fraction vector [%d] does not equal that of component vector [%d]",mole_fractions.size(), N));
    }
    // Copy values without reallocating memory
    this->mole_fractions = mole_fractions; // Most effective copy
    this->resize(N); // No reallocation of this->mole_fractions happens
    // Also store the mole fractions as doubles
    this->mole_fractions_double = std::vector<double>(mole_fractions.begin(), mole_fractions.end());
};

void PCSAFTBackend::set_mass_fractions(const std::vector<CoolPropDbl> &mass_fractions)
{
    if (mass_fractions.size() != N)
    {
        throw ValueError(format("size of mass fraction vector [%d] does not equal that of component vector [%d]",mass_fractions.size(), N));
    }
    std::vector<CoolPropDbl> moles;
	CoolPropDbl sum_moles = 0.0;
	CoolPropDbl tmp = 0.0;
    for (unsigned int i = 0; i < components.size(); ++i)
    {
        tmp = mass_fractions[i]/components[i].molar_mass();
        moles.push_back(tmp);
		sum_moles += tmp;
    }
	std::vector<CoolPropDbl> mole_fractions;
	for(std::vector< CoolPropDbl >::iterator it = moles.begin(); it != moles.end(); ++it)
    {
		mole_fractions.push_back(*it/sum_moles);
	}
	this->set_mole_fractions(mole_fractions);
};

PCSAFTBackend * PCSAFTBackend::get_copy(bool generate_SatL_and_SatV){
    // Set up the class with these components
    PCSAFTBackend * ptr = new PCSAFTBackend(components, generate_SatL_and_SatV);
    return ptr;
};

void PCSAFTBackend::resize(std::size_t N)
{
    this->mole_fractions.resize(N);
    this->mole_fractions_double.resize(N);
    this->K.resize(N);
    this->lnK.resize(N);
}

CoolPropDbl PCSAFTBackend::update_DmolarT(CoolPropDbl rho) {
    _rhomolar = rho;
    return this->calc_pressure();
}

CoolPropDbl PCSAFTBackend::calc_pressure(void) {
    double den = _rhomolar*N_AV/1.0e30;

    CoolPropDbl Z = this->calc_compressibility_factor();
    CoolPropDbl P = Z*kb*_T*den*1.0e30; // Pa
    return P;
}

CoolPropDbl PCSAFTBackend::calc_alphar(void) {
    int ncomp = N; // number of components
    vector<double> d (ncomp);
    for (int i = 0; i < ncomp; i++) {
        d[i] = components[i].getSigma()*(1-0.12*exp(-3*components[i].getU()/_T));
    }
    if (ion_term) {
        for (int i = 0; i < ncomp; i++) {
            if (components[i].getZ() != 0) {
                d[i] = components[i].getSigma()*(1-0.12); // for ions the diameter is assumed to be temperature independent (see Held et al. 2014)
            }
        }
    }

    double den = _rhomolar*N_AV/1.0e30;

    vector<double> zeta (4, 0);
    double summ;
    for (int i = 0; i < 4; i++) {
        summ = 0;
        for (int j = 0; j < ncomp; j++) {
            summ += mole_fractions[j]*components[j].getM()*pow(d[j], i);
        }
        zeta[i] = PI/6*den*summ;
    }

    double eta = zeta[3];
    double m_avg = 0;
    for (int i = 0; i < ncomp; i++) {
        m_avg += mole_fractions[i]*components[i].getM();
    }

    vector<double> ghs (ncomp*ncomp, 0);
    vector<double> e_ij (ncomp*ncomp, 0);
    vector<double> s_ij (ncomp*ncomp, 0);
    double m2es3 = 0.;
    double m2e2s3 = 0.;
    int idx = -1;
    for (int i = 0; i < ncomp; i++) {
        for (int j = 0; j < ncomp; j++) {
            idx += 1;
            s_ij[idx] = (components[i].getSigma() + components[j].getSigma())/2.;
            if (ion_term) {
                if (components[i].getZ()*components[j].getZ() <= 0) { // for two cations or two anions e_ij is kept at zero to avoid dispersion between like ions (see Held et al. 2014)
                    if (k_ij.empty()) {
                        e_ij[idx] = sqrt(components[i].getU()*components[j].getU());
                    }
                    else {
                        e_ij[idx] = sqrt(components[i].getU()*components[j].getU())*(1 - (k_ij[idx] + k_ijT[idx] * _T));
                    }
                }
            } else {
                if (k_ij.empty()) {
                    e_ij[idx] = sqrt(components[i].getU()*components[j].getU());
                }
                else {
                    e_ij[idx] = sqrt(components[i].getU()*components[j].getU())*(1 - (k_ij[idx] + k_ijT[idx] * _T));
                }
            }
            m2es3 = m2es3 + mole_fractions[i]*mole_fractions[j]*components[i].getM()*components[j].getM()*e_ij[idx]/_T*pow(s_ij[idx], 3);
            m2e2s3 = m2e2s3 + mole_fractions[i]*mole_fractions[j]*components[i].getM()*components[j].getM()*pow(e_ij[idx]/_T,2)*pow(s_ij[idx], 3);
            ghs[idx] = 1/(1-zeta[3]) + (d[i]*d[j]/(d[i]+d[j]))*3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
                pow(d[i]*d[j]/(d[i]+d[j]), 2)*2*zeta[2]*zeta[2]/pow(1-zeta[3], 3);
        }
    }

    double ares_hs = 1/zeta[0]*(3*zeta[1]*zeta[2]/(1-zeta[3]) + pow(zeta[2], 3.)/(zeta[3]*pow(1-zeta[3],2))
            + (pow(zeta[2], 3.)/pow(zeta[3], 2.) - zeta[0])*log(1-zeta[3]));

    static double a0[7] = { 0.910563145, 0.636128145, 2.686134789, -26.54736249, 97.75920878, -159.5915409, 91.29777408 };
    static double a1[7] = { -0.308401692, 0.186053116, -2.503004726, 21.41979363, -65.25588533, 83.31868048, -33.74692293 };
    static double a2[7] = { -0.090614835, 0.452784281, 0.596270073, -1.724182913, -4.130211253, 13.77663187, -8.672847037 };
    static double b0[7] = { 0.724094694, 2.238279186, -4.002584949, -21.00357682, 26.85564136, 206.5513384, -355.6023561 };
    static double b1[7] = { -0.575549808, 0.699509552, 3.892567339, -17.21547165, 192.6722645, -161.8264617, -165.2076935 };
    static double b2[7] = { 0.097688312, -0.255757498, -9.155856153, 20.64207597, -38.80443005, 93.62677408, -29.66690559 };

    vector<double> a (7, 0);
    vector<double> b (7, 0);
    for (int i = 0; i < 7; i++) {
        a[i] = a0[i] + (m_avg-1.)/m_avg*a1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*a2[i];
        b[i] = b0[i] + (m_avg-1.)/m_avg*b1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*b2[i];
    }

    double I1 = 0.0;
    double I2 = 0.0;
    for (int i = 0; i < 7; i++) {
        I1 += a[i]*pow(eta, i);
        I2 += b[i]*pow(eta, i);
    }
    double C1 = 1./(1. + m_avg*(8*eta-2*eta*eta)/pow(1-eta, 4) + (1-m_avg)*(20*eta-27*eta*eta+12*pow(eta, 3)-2*pow(eta, 4))/pow((1-eta)*(2-eta), 2.0));

    summ = 0.0;
    for (int i = 0; i < ncomp; i++) {
        summ += mole_fractions[i]*(components[i].getM()-1)*log(ghs[i*ncomp+i]);
    }

    double ares_hc = m_avg*ares_hs - summ;
    double ares_disp = -2*PI*den*I1*m2es3 - PI*den*m_avg*C1*I2*m2e2s3;

    // Dipole term (Gross and Vrabec term) --------------------------------------
    double ares_polar = 0.;
    if (polar_term) {
        double A2 = 0.;
        double A3 = 0.;
        vector<double> dipmSQ (ncomp, 0);

        static double a0dip[5] = { 0.3043504, -0.1358588, 1.4493329, 0.3556977, -2.0653308 };
        static double a1dip[5] = { 0.9534641, -1.8396383, 2.0131180, -7.3724958, 8.2374135 };
        static double a2dip[5] = { -1.1610080, 4.5258607, 0.9751222, -12.281038, 5.9397575 };
        static double b0dip[5] = { 0.2187939, -1.1896431, 1.1626889, 0, 0 };
        static double b1dip[5] = { -0.5873164, 1.2489132, -0.5085280, 0, 0 };
        static double b2dip[5] = { 3.4869576, -14.915974, 15.372022, 0, 0 };
        static double c0dip[5] = { -0.0646774, 0.1975882, -0.8087562, 0.6902849, 0 };
        static double c1dip[5] = { -0.9520876, 2.9924258, -2.3802636, -0.2701261, 0 };
        static double c2dip[5] = { -0.6260979, 1.2924686, 1.6542783, -3.4396744, 0 };

        const static double conv = 7242.702976750923; // conversion factor, see the note below Table 2 in Gross and Vrabec 2006

        for (int i = 0; i < ncomp; i++) {
            dipmSQ[i] = pow(components[i].getDipm(), 2.)/(components[i].getM()*components[i].getU()*pow(components[i].getSigma(),3.))*conv;
        }

        vector<double> adip (5, 0);
        vector<double> bdip (5, 0);
        vector<double> cdip (5, 0);
        double J2, J3;
        double m_ij;
        double m_ijk;
        for (int i = 0; i < ncomp; i++) {
            for (int j = 0; j < ncomp; j++) {
                m_ij = sqrt(components[i].getM()*components[j].getM());
                if (m_ij > 2) {
                    m_ij = 2;
                }
                J2 = 0.;
                for (int l = 0; l < 5; l++) {
                    adip[l] = a0dip[l] + (m_ij-1)/m_ij*a1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*a2dip[l];
                    bdip[l] = b0dip[l] + (m_ij-1)/m_ij*b1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*b2dip[l];
                    J2 += (adip[l] + bdip[l]*e_ij[j*ncomp+j]/_T)*pow(eta, l); // j*ncomp+j needs to be used for e_ij because it is formatted as a 1D vector
                }
                A2 += mole_fractions[i]*mole_fractions[j]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)/
                    pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*J2;

                for (int k = 0; k < ncomp; k++) {
                    m_ijk = pow((components[i].getM()*components[j].getM()*components[k].getM()),1/3.);
                    if (m_ijk > 2) {
                        m_ijk = 2;
                    }
                    J3 = 0.;
                    for (int l = 0; l < 5; l++) {
                        cdip[l] = c0dip[l] + (m_ijk-1)/m_ijk*c1dip[l] + (m_ijk-1)/m_ijk*(m_ijk-2)/m_ijk*c2dip[l];
                        J3 += cdip[l]*pow(eta, l);
                    }
                    A3 += mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*
                        pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
                        s_ij[j*ncomp+k]*components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*
                        dipmSQ[j]*dipmSQ[k]*J3;
                }
            }
        }

        A2 = -PI*den*A2;
        A3 = -4/3.*PI*PI*den*den*A3;

        ares_polar = A2/(1-A3/A2);
    }

    // Association term -------------------------------------------------------
    // only the 2B association type is currently implemented
    double ares_assoc = 0.;
    if (assoc_term) {
        int a_sites = 2;
        int ncA = 0; // number of associating compounds
        vector<int> iA; // indices of associating compounds
        for (int i = 0; i < ncomp; i++) {
            if (components[i].getVolA() != 0) {
                iA.push_back(i);
                ncA += 1;
            }
        }

        vector<double> XA (ncA*a_sites, 0);
        vector<double> eABij (ncA*ncA, 0);
        vector<double> volABij (ncA*ncA, 0);
        vector<double> delta_ij (ncA*ncA, 0);

        // these indices are necessary because we are only using 1D vectors
        int idxa = -1; // index over only associating compounds
        int idxi = 0; // index for the ii-th compound
        int idxj = 0; // index for the jj-th compound
        for (int i = 0; i < ncA; i++) {
            idxi = iA[i]*ncomp+iA[i];
            for (int j = 0; j < ncA; j++) {
                idxa += 1;
                idxj = iA[j]*ncomp+iA[j];
                eABij[idxa] = (components[iA[i]].getUAB()+components[iA[j]].getUAB())/2.;
                volABij[idxa] = sqrt(components[iA[i]].getVolA()*components[iA[j]].getVolA())*pow(sqrt(s_ij[idxi]*
                    s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3);
                delta_ij[idxa] = ghs[iA[i]*ncomp+iA[j]]*(exp(eABij[idxa]/_T)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij[idxa];
            }
            XA[i*2] = (-1 + sqrt(1+8*den*delta_ij[i*ncA+i]))/(4*den*delta_ij[i*ncA+i]);
            if (!ValidNumber(XA[i*2])) {
                XA[i*2] = 0.02;
            }
            XA[i*2+1] = XA[i*2];
        }

        vector<double> x_assoc(ncA); // mole fractions of only the associating compounds
        for (int i = 0; i < ncA; i++) {
            x_assoc[i] = mole_fractions[iA[i]];
        }

        int ctr = 0;
        double dif = 1000.;
        vector<double> XA_old = XA;
        while ((ctr < 500) && (dif > 1e-9)) {
            ctr += 1;
            XA = XA_find(XA, ncA, delta_ij, den, x_assoc);
            dif = 0.;
            for (int i = 0; i < ncA*2; i++) {
                dif += abs(XA[i] - XA_old[i]);
            }
            XA_old = XA;
        }

        ares_assoc = 0.;
        for (int i = 0; i < ncA; i++) {
            for (int k = 0; k < a_sites; k++) {
                ares_assoc += mole_fractions[iA[i]]*(log(XA[i*a_sites+k])-0.5*XA[i*a_sites+k] + 0.5);
            }
        }
    }

    // Ion term ---------------------------------------------------------------
    double ares_ion = 0.;
    if (ion_term) {
        vector<double> q(ncomp);
        for (int i = 0; i < ncomp; i++) {
            q[i] = components[i].getZ()*E_CHRG;
        }

        summ = 0.;
        for (int i = 0; i < ncomp; i++) {
            summ += components[i].getZ()*components[i].getZ()*mole_fractions[i];
        }
        double kappa = sqrt(den*E_CHRG*E_CHRG/kb/_T/(dielc*perm_vac)*summ); // the inverse Debye screening length. Equation 4 in Held et al. 2008.

        if (kappa != 0) {
            vector<double> chi(ncomp);
            vector<double> sigma_k(ncomp);
            summ = 0.;
            for (int i = 0; i < ncomp; i++) {
                chi[i] = 3/pow(kappa*components[i].getSigma(), 3)*(1.5 + log(1+kappa*components[i].getSigma()) - 2*(1+kappa*components[i].getSigma()) +
                    0.5*pow(1+kappa*components[i].getSigma(), 2));
                summ += mole_fractions[i]*q[i]*q[i]*chi[i]*kappa;
            }

            ares_ion = -1/12./PI/kb/_T/(dielc*perm_vac)*summ;
        }
    }

    CoolPropDbl ares = ares_hc + ares_disp + ares_polar + ares_assoc + ares_ion;
    return ares;
}

CoolPropDbl PCSAFTBackend::calc_dadt(void) {
    int ncomp = N; // number of components
    vector<double> d (ncomp), dd_dt(ncomp);
    for (int i = 0; i < ncomp; i++) {
        d[i] = components[i].getSigma()*(1-0.12*exp(-3*components[i].getU()/_T));
        dd_dt[i] = components[i].getSigma()*-3*components[i].getU()/_T/_T*0.12*exp(-3*components[i].getU()/_T);
    }
    if (ion_term) {
        for (int i = 0; i < ncomp; i++) {
            if (components[i].getZ() != 0) {
                d[i] = components[i].getSigma()*(1-0.12); // for ions the diameter is assumed to be temperature independent (see Held et al. 2014)
                dd_dt[i] = 0.;
            }
        }
    }

    double den = _rhomolar*N_AV/1.0e30;

    vector<double> zeta (4, 0);
    double summ;
    for (int i = 0; i < 4; i++) {
        summ = 0;
        for (int j = 0; j < ncomp; j++) {
            summ += mole_fractions[j]*components[j].getM()*pow(d[j], i);
        }
        zeta[i] = PI/6*den*summ;
    }

    vector<double> dzeta_dt (4, 0);
    for (int i = 1; i < 4; i++) {
        summ = 0;
        for (int j = 0; j < ncomp; j++) {
            summ += mole_fractions[j]*components[j].getM()*i*dd_dt[j]*pow(d[j],(i-1));
        }
        dzeta_dt[i] = PI/6*den*summ;
    }

    double eta = zeta[3];
    double m_avg = 0;
    for (int i = 0; i < ncomp; i++) {
        m_avg += mole_fractions[i]*components[i].getM();
    }

    vector<double> ghs (ncomp*ncomp, 0);
    vector<double> dghs_dt (ncomp*ncomp, 0);
    vector<double> e_ij (ncomp*ncomp, 0);
    vector<double> s_ij (ncomp*ncomp, 0);
    double m2es3 = 0.;
    double m2e2s3 = 0.;
    double ddij_dt;
    int idx = -1;
    for (int i = 0; i < ncomp; i++) {
        for (int j = 0; j < ncomp; j++) {
            idx += 1;
            s_ij[idx] = (components[i].getSigma() + components[j].getSigma())/2.;
            if (ion_term) {
                if (components[i].getZ()*components[j].getZ() <= 0) { // for two cations or two anions e_ij is kept at zero to avoid dispersion between like ions (see Held et al. 2014)
                    if (k_ij.empty()) {
                        e_ij[idx] = sqrt(components[i].getU()*components[j].getU());
                    }
                    else {
                        e_ij[idx] = sqrt(components[i].getU()*components[j].getU())*(1 - (k_ij[idx] + k_ijT[idx] * _T));
                    }
                }
            } else {
                if (k_ij.empty()) {
                    e_ij[idx] = sqrt(components[i].getU()*components[j].getU());
                }
                else {
                    e_ij[idx] = sqrt(components[i].getU()*components[j].getU())*(1 - (k_ij[idx] + k_ijT[idx] * _T));
                }
            }
            m2es3 = m2es3 + mole_fractions[i]*mole_fractions[j]*components[i].getM()*components[j].getM()*e_ij[idx]/_T*pow(s_ij[idx], 3);
            m2e2s3 = m2e2s3 + mole_fractions[i]*mole_fractions[j]*components[i].getM()*components[j].getM()*pow(e_ij[idx]/_T,2)*pow(s_ij[idx], 3);
            ghs[idx] = 1/(1-zeta[3]) + (d[i]*d[j]/(d[i]+d[j]))*3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
                    pow(d[i]*d[j]/(d[i]+d[j]), 2)*2*zeta[2]*zeta[2]/pow(1-zeta[3], 3);
            ddij_dt = (d[i]*d[j]/(d[i]+d[j]))*(dd_dt[i]/d[i]+dd_dt[j]/d[j]-(dd_dt[i]+dd_dt[j])/(d[i]+d[j]));
            dghs_dt[idx] = dzeta_dt[3]/pow(1-zeta[3], 2.)
                + 3*(ddij_dt*zeta[2]+(d[i]*d[j]/(d[i]+d[j]))*dzeta_dt[2])/pow(1-zeta[3], 2.)
                + 4*(d[i]*d[j]/(d[i]+d[j]))*zeta[2]*(1.5*dzeta_dt[3]+ddij_dt*zeta[2]
                + (d[i]*d[j]/(d[i]+d[j]))*dzeta_dt[2])/pow(1-zeta[3], 3.)
                + 6*pow((d[i]*d[j]/(d[i]+d[j]))*zeta[2], 2.)*dzeta_dt[3]/pow(1-zeta[3], 4.);
        }
    }

    double dadt_hs = 1/zeta[0]*(3*(dzeta_dt[1]*zeta[2] + zeta[1]*dzeta_dt[2])/(1-zeta[3])
        + 3*zeta[1]*zeta[2]*dzeta_dt[3]/pow(1-zeta[3], 2.)
        + 3*pow(zeta[2], 2.)*dzeta_dt[2]/zeta[3]/pow(1-zeta[3], 2.)
        + pow(zeta[2],3.)*dzeta_dt[3]*(3*zeta[3]-1)/pow(zeta[3], 2.)/pow(1-zeta[3], 3.)
        + (3*pow(zeta[2], 2.)*dzeta_dt[2]*zeta[3] - 2*pow(zeta[2], 3.)*dzeta_dt[3])/pow(zeta[3], 3.)
        * log(1-zeta[3])
        + (zeta[0]-pow(zeta[2],3)/pow(zeta[3],2.))*dzeta_dt[3]/(1-zeta[3]));

    static double a0[7] = { 0.910563145, 0.636128145, 2.686134789, -26.54736249, 97.75920878, -159.5915409, 91.29777408 };
    static double a1[7] = { -0.308401692, 0.186053116, -2.503004726, 21.41979363, -65.25588533, 83.31868048, -33.74692293 };
    static double a2[7] = { -0.090614835, 0.452784281, 0.596270073, -1.724182913, -4.130211253, 13.77663187, -8.672847037 };
    static double b0[7] = { 0.724094694, 2.238279186, -4.002584949, -21.00357682, 26.85564136, 206.5513384, -355.6023561 };
    static double b1[7] = { -0.575549808, 0.699509552, 3.892567339, -17.21547165, 192.6722645, -161.8264617, -165.2076935 };
    static double b2[7] = { 0.097688312, -0.255757498, -9.155856153, 20.64207597, -38.80443005, 93.62677408, -29.66690559 };

    vector<double> a (7, 0);
    vector<double> b (7, 0);
    for (int i = 0; i < 7; i++) {
        a[i] = a0[i] + (m_avg-1.)/m_avg*a1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*a2[i];
        b[i] = b0[i] + (m_avg-1.)/m_avg*b1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*b2[i];
    }

    double I1 = 0.0;
    double I2 = 0.0;
    double dI1_dt = 0.0, dI2_dt = 0.;
    for (int i = 0; i < 7; i++) {
        I1 += a[i]*pow(eta, i);
        I2 += b[i]*pow(eta, i);
        dI1_dt += a[i]*dzeta_dt[3]*i*pow(eta, i-1);
        dI2_dt += b[i]*dzeta_dt[3]*i*pow(eta, i-1);
    }
    double C1 = 1./(1. + m_avg*(8*eta-2*eta*eta)/pow(1-eta, 4) + (1-m_avg)*(20*eta-27*eta*eta+12*pow(eta, 3)-2*pow(eta, 4))/pow((1-eta)*(2-eta), 2.0));
    double C2 = -1*C1*C1*(m_avg*(-4*eta*eta+20*eta+8)/pow(1-eta,5.) + (1-m_avg)*(2*pow(eta,3)+12*eta*eta-48*eta+40)/pow((1-eta)*(2-eta),3));
    double dC1_dt = C2*dzeta_dt[3];

    summ = 0.;
    for (int i = 0; i < ncomp; i++) {
        summ += mole_fractions[i]*(components[i].getM()-1)*dghs_dt[i*ncomp+i]/ghs[i*ncomp+i];
    }

    double dadt_hc = m_avg*dadt_hs - summ;
    double dadt_disp = -2*PI*den*(dI1_dt-I1/_T)*m2es3 - PI*den*m_avg*(dC1_dt*I2+C1*dI2_dt-2*C1*I2/_T)*m2e2s3;

    // Dipole term (Gross and Vrabec term) --------------------------------------
    double dadt_polar = 0.;
    if (polar_term) {
        double A2 = 0.;
        double A3 = 0.;
        double dA2_dt = 0.;
        double dA3_dt = 0.;
        vector<double> dipmSQ (ncomp, 0);

        static double a0dip[5] = { 0.3043504, -0.1358588, 1.4493329, 0.3556977, -2.0653308 };
        static double a1dip[5] = { 0.9534641, -1.8396383, 2.0131180, -7.3724958, 8.2374135 };
        static double a2dip[5] = { -1.1610080, 4.5258607, 0.9751222, -12.281038, 5.9397575 };
        static double b0dip[5] = { 0.2187939, -1.1896431, 1.1626889, 0, 0 };
        static double b1dip[5] = { -0.5873164, 1.2489132, -0.5085280, 0, 0 };
        static double b2dip[5] = { 3.4869576, -14.915974, 15.372022, 0, 0 };
        static double c0dip[5] = { -0.0646774, 0.1975882, -0.8087562, 0.6902849, 0 };
        static double c1dip[5] = { -0.9520876, 2.9924258, -2.3802636, -0.2701261, 0 };
        static double c2dip[5] = { -0.6260979, 1.2924686, 1.6542783, -3.4396744, 0 };

        const static double conv = 7242.702976750923; // conversion factor, see the note below Table 2 in Gross and Vrabec 2006

        for (int i = 0; i < ncomp; i++) {
            dipmSQ[i] = pow(components[i].getDipm(), 2.)/(components[i].getM()*components[i].getU()*pow(components[i].getSigma(),3.))*conv;
        }

        vector<double> adip (5, 0);
        vector<double> bdip (5, 0);
        vector<double> cdip (5, 0);
        double J2, J3, dJ2_dt, dJ3_dt;
        double m_ij;
        double m_ijk;
        for (int i = 0; i < ncomp; i++) {
            for (int j = 0; j < ncomp; j++) {
                m_ij = sqrt(components[i].getM()*components[j].getM());
                if (m_ij > 2) {
                    m_ij = 2;
                }
                J2 = 0.;
                dJ2_dt = 0.;
                for (int l = 0; l < 5; l++) {
                    adip[l] = a0dip[l] + (m_ij-1)/m_ij*a1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*a2dip[l];
                    bdip[l] = b0dip[l] + (m_ij-1)/m_ij*b1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*b2dip[l];
                    J2 += (adip[l] + bdip[l]*e_ij[j*ncomp+j]/_T)*pow(eta, l); // j*ncomp+j needs to be used for e_ij because it is formatted as a 1D vector
                    dJ2_dt += adip[l]*l*pow(eta, l-1)*dzeta_dt[3]
                        + bdip[l]*e_ij[j*ncomp+j]*(1/_T*l*pow(eta, l-1)*dzeta_dt[3]
                        - 1/pow(_T,2.)*pow(eta,l));
                }
                A2 += mole_fractions[i]*mole_fractions[j]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)/
                    pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*J2;
                dA2_dt += mole_fractions[i]*mole_fractions[j]*e_ij[i*ncomp+i]*e_ij[j*ncomp+j]*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)
                    /pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*
                    (dJ2_dt/pow(_T,2)-2*J2/pow(_T,3));

                for (int k = 0; k < ncomp; k++) {
                    m_ijk = pow((components[i].getM()*components[j].getM()*components[k].getM()),1/3.);
                    if (m_ijk > 2) {
                        m_ijk = 2;
                    }
                    J3 = 0.;
                    dJ3_dt = 0.;
                    for (int l = 0; l < 5; l++) {
                        cdip[l] = c0dip[l] + (m_ijk-1)/m_ijk*c1dip[l] + (m_ijk-1)/m_ijk*(m_ijk-2)/m_ijk*c2dip[l];
                        J3 += cdip[l]*pow(eta, l);
                        dJ3_dt += cdip[l]*l*pow(eta, l-1)*dzeta_dt[3];
                    }
                    A3 += mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*
                        pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
                        s_ij[j*ncomp+k]*components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*
                        dipmSQ[j]*dipmSQ[k]*J3;
                    dA3_dt += mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*e_ij[i*ncomp+i]*e_ij[j*ncomp+j]*e_ij[k*ncomp+k]*
                        pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]
                        /s_ij[j*ncomp+k]*components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]
                        *dipmSQ[j]*dipmSQ[k]*(-3*J3/pow(_T,4) + dJ3_dt/pow(_T,3));
                }
            }
        }

        A2 = -PI*den*A2;
        A3 = -4/3.*PI*PI*den*den*A3;
        dA2_dt = -PI*den*dA2_dt;
        dA3_dt = -4/3.*PI*PI*den*den*dA3_dt;

        dadt_polar = (dA2_dt-2*A3/A2*dA2_dt+dA3_dt)/pow(1-A3/A2, 2.);
    }

    // Association term -------------------------------------------------------
    // only the 2B association type is currently implemented
    double dadt_assoc = 0.;
    if (assoc_term) {
        int a_sites = 2;
        int ncA = 0; // number of associating compounds
        vector<int> iA; // indices of associating compounds
        for (int i = 0; i < ncomp; i++) {
            if (components[i].getVolA() != 0) {
                iA.push_back(i);
                ncA += 1;
            }
        }

        vector<double> XA (ncA*a_sites, 0);
        vector<double> eABij (ncA*ncA, 0);
        vector<double> volABij (ncA*ncA, 0);
        vector<double> delta_ij (ncA*ncA, 0);
        vector<double> ddelta_dt (ncA*ncA, 0);

        // these indices are necessary because we are only using 1D vectors
        int idxa = -1; // index over only associating compounds
        int idxi = 0; // index for the ii-th compound
        int idxj = 0; // index for the jj-th compound
        for (int i = 0; i < ncA; i++) {
            idxi = iA[i]*ncomp+iA[i];
            for (int j = 0; j < ncA; j++) {
                idxa += 1;
                idxj = iA[j]*ncomp+iA[j];
                eABij[idxa] = (components[iA[i]].getUAB()+components[iA[j]].getUAB())/2.;
                volABij[idxa] = sqrt(components[iA[i]].getVolA()*components[iA[j]].getVolA())*pow(sqrt(s_ij[idxi]*
                    s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3);
                delta_ij[idxa] = ghs[iA[i]*ncomp+iA[j]]*(exp(eABij[idxa]/_T)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij[idxa];
                ddelta_dt[idxa] = pow(s_ij[idxj],3)*volABij[idxa]*(-eABij[idxa]/pow(_T,2)
                    *exp(eABij[idxa]/_T)*ghs[iA[i]*ncomp+iA[j]] + dghs_dt[iA[i]*ncomp+iA[j]]
                    *(exp(eABij[idxa]/_T)-1));
            }
            XA[i*2] = (-1 + sqrt(1+8*den*delta_ij[i*ncA+i]))/(4*den*delta_ij[i*ncA+i]);
            if (!ValidNumber(XA[i*2])) {
                XA[i*2] = 0.02;
            }
            XA[i*2+1] = XA[i*2];
        }

        vector<double> x_assoc(ncA); // mole fractions of only the associating compounds
        for (int i = 0; i < ncA; i++) {
            x_assoc[i] = mole_fractions[iA[i]];
        }

        int ctr = 0;
        double dif = 1000.;
        vector<double> XA_old = XA;
        while ((ctr < 500) && (dif > 1e-9)) {
            ctr += 1;
            XA = XA_find(XA, ncA, delta_ij, den, x_assoc);
            dif = 0.;
            for (int i = 0; i < ncA*2; i++) {
                dif += abs(XA[i] - XA_old[i]);
            }
            XA_old = XA;
        }

        vector<double> dXA_dt (ncA*a_sites, 0);
        dXA_dt = dXAdt_find(ncA, delta_ij, den, XA, ddelta_dt, x_assoc, a_sites);

        int idx = -1;
        for (int i = 0; i < ncA; i++) {
            for (int j = 0; j < a_sites; j++) {
                idx += 1;
                dadt_assoc += mole_fractions[iA[i]]*(1/XA[idx]-0.5)*dXA_dt[idx];
            }
        }
    }

    // Ion term ---------------------------------------------------------------
    double dadt_ion = 0.;
    if (ion_term) {
        vector<double> q(ncomp);
        for (int i = 0; i < ncomp; i++) {
            q[i] = components[i].getZ()*E_CHRG;
        }

        summ = 0.;
        for (int i = 0; i < ncomp; i++) {
            summ += components[i].getZ()*components[i].getZ()*mole_fractions[i];
        }
        double kappa = sqrt(den*E_CHRG*E_CHRG/kb/_T/(dielc*perm_vac)*summ); // the inverse Debye screening length. Equation 4 in Held et al. 2008.

        double dkappa_dt;
        if (kappa != 0) {
            vector<double> chi(ncomp);
            vector<double> dchikap_dk(ncomp);
            summ = 0.;
            for (int i = 0; i < ncomp; i++) {
                chi[i] = 3/pow(kappa*components[i].getSigma(), 3)*(1.5 + log(1+kappa*components[i].getSigma()) - 2*(1+kappa*components[i].getSigma()) +
                    0.5*pow(1+kappa*components[i].getSigma(), 2));
                dchikap_dk[i] = -2*chi[i]+3/(1+kappa*components[i].getSigma());
                summ += mole_fractions[i]*components[i].getZ()*components[i].getZ();
            }
            dkappa_dt = -0.5*den*E_CHRG*E_CHRG/kb/_T/_T/(dielc*perm_vac)*summ/kappa;

            summ = 0.;
            for (int i = 0; i < ncomp; i++) {
                summ += mole_fractions[i]*q[i]*q[i]*(dchikap_dk[i]*dkappa_dt/_T-kappa*chi[i]/_T/_T);
            }
            dadt_ion = -1/12./PI/kb/(dielc*perm_vac)*summ;
        }
    }

    double dadt = dadt_hc + dadt_disp + dadt_assoc + dadt_polar + dadt_ion;
    return dadt;
}

CoolPropDbl PCSAFTBackend::calc_hmolar_residual(void) {
    CoolPropDbl Z = calc_compressibility_factor();
    CoolPropDbl dares_dt = calc_dadt();

    CoolPropDbl hres = (-_T*dares_dt + (Z-1))*kb*N_AV*_T; // Equation A.46 from Gross and Sadowski 2001
    return hres;
}

CoolPropDbl PCSAFTBackend::calc_smolar_residual(void){
    CoolPropDbl dares_dt = calc_dadt();
    CoolPropDbl ares = calc_alphar();

    CoolPropDbl sres = kb * N_AV * (-_T * dares_dt - ares);
    return sres;
}

vector<CoolPropDbl> PCSAFTBackend::calc_fugacity_coefficients(void) {
    int ncomp = N; // number of components
    vector<double> d(ncomp);
    for (int i = 0; i < ncomp; i++) {
       d[i] = components[i].getSigma()*(1-0.12*exp(-3*components[i].getU()/_T));
    }
    if (ion_term) {
       for (int i = 0; i < ncomp; i++) {
           if (components[i].getZ() != 0) {
               d[i] = components[i].getSigma()*(1-0.12); // for ions the diameter is assumed to be temperature independent (see Held et al. 2014)
           }
       }
    }

    double den = _rhomolar*N_AV/1.0e30;

    vector<double> zeta (4, 0);
    double summ;
    for (int i = 0; i < 4; i++) {
       summ = 0;
       for (int j = 0; j < ncomp; j++) {
           summ += mole_fractions[j]*components[j].getM()*pow(d[j], i);
       }
       zeta[i] = PI/6*den*summ;
    }

    double eta = zeta[3];
    double m_avg = 0;
    for (int i = 0; i < ncomp; i++) {
       m_avg += mole_fractions[i]*components[i].getM();
    }

    vector<double> ghs(ncomp*ncomp, 0);
    vector<double> denghs(ncomp*ncomp, 0);
    vector<double> e_ij(ncomp*ncomp, 0);
    vector<double> s_ij(ncomp*ncomp, 0);
    double m2es3 = 0.;
    double m2e2s3 = 0.;
    int idx = -1;
    for (int i = 0; i < ncomp; i++) {
       for (int j = 0; j < ncomp; j++) {
           idx += 1;
           s_ij[idx] = (components[i].getSigma() + components[j].getSigma())/2.;
           if (ion_term) {
               if (components[i].getZ()*components[j].getZ() <= 0) { // for two cations or two anions e_ij is kept at zero to avoid dispersion between like ions (see Held et al. 2014)
                   if (k_ij.empty()) {
                       e_ij[idx] = sqrt(components[i].getU()*components[j].getU());
                   }
                   else {
                       e_ij[idx] = sqrt(components[i].getU()*components[j].getU())*(1 - (k_ij[idx] + k_ijT[idx] * _T));
                   }
               }
           } else {
               if (k_ij.empty()) {
                   e_ij[idx] = sqrt(components[i].getU()*components[j].getU());
               }
               else {
                   e_ij[idx] = sqrt(components[i].getU()*components[j].getU())*(1 - (k_ij[idx] + k_ijT[idx] * _T));
               }
           }
           m2es3 = m2es3 + mole_fractions[i]*mole_fractions[j]*components[i].getM()*components[j].getM()*e_ij[idx]/_T*pow(s_ij[idx], 3);
           m2e2s3 = m2e2s3 + mole_fractions[i]*mole_fractions[j]*components[i].getM()*components[j].getM()*pow(e_ij[idx]/_T,2)*pow(s_ij[idx], 3);
           ghs[idx] = 1/(1-zeta[3]) + (d[i]*d[j]/(d[i]+d[j]))*3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
                   pow(d[i]*d[j]/(d[i]+d[j]), 2)*2*zeta[2]*zeta[2]/pow(1-zeta[3], 3);
           denghs[idx] = zeta[3]/(1-zeta[3])/(1-zeta[3]) +
               (d[i]*d[j]/(d[i]+d[j]))*(3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
               6*zeta[2]*zeta[3]/pow(1-zeta[3], 3)) +
               pow(d[i]*d[j]/(d[i]+d[j]), 2)*(4*zeta[2]*zeta[2]/pow(1-zeta[3], 3) +
               6*zeta[2]*zeta[2]*zeta[3]/pow(1-zeta[3], 4));
       }
    }

    double ares_hs = 1/zeta[0]*(3*zeta[1]*zeta[2]/(1-zeta[3]) + pow(zeta[2], 3.)/(zeta[3]*pow(1-zeta[3],2))
           + (pow(zeta[2], 3.)/pow(zeta[3], 2.) - zeta[0])*log(1-zeta[3]));
    double Zhs = zeta[3]/(1-zeta[3]) + 3.*zeta[1]*zeta[2]/zeta[0]/(1.-zeta[3])/(1.-zeta[3]) +
       (3.*pow(zeta[2], 3.) - zeta[3]*pow(zeta[2], 3.))/zeta[0]/pow(1.-zeta[3], 3.);

    static double a0[7] = { 0.910563145, 0.636128145, 2.686134789, -26.54736249, 97.75920878, -159.5915409, 91.29777408 };
    static double a1[7] = { -0.308401692, 0.186053116, -2.503004726, 21.41979363, -65.25588533, 83.31868048, -33.74692293 };
    static double a2[7] = { -0.090614835, 0.452784281, 0.596270073, -1.724182913, -4.130211253, 13.77663187, -8.672847037 };
    static double b0[7] = { 0.724094694, 2.238279186, -4.002584949, -21.00357682, 26.85564136, 206.5513384, -355.6023561 };
    static double b1[7] = { -0.575549808, 0.699509552, 3.892567339, -17.21547165, 192.6722645, -161.8264617, -165.2076935 };
    static double b2[7] = { 0.097688312, -0.255757498, -9.155856153, 20.64207597, -38.80443005, 93.62677408, -29.66690559 };

    vector<double> a (7, 0);
    vector<double> b (7, 0);
    for (int i = 0; i < 7; i++) {
       a[i] = a0[i] + (m_avg-1.)/m_avg*a1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*a2[i];
       b[i] = b0[i] + (m_avg-1.)/m_avg*b1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*b2[i];
    }

    double detI1_det = 0.0;
    double detI2_det = 0.0;
    double I1 = 0.0;
    double I2 = 0.0;
    for (int i = 0; i < 7; i++) {
       detI1_det += a[i]*(i+1)*pow(eta, i);
       detI2_det += b[i]*(i+1)*pow(eta, i);
       I2 += b[i]*pow(eta, i);
       I1 += a[i]*pow(eta, i);
    }
    double C1 = 1./(1. + m_avg*(8*eta-2*eta*eta)/pow(1-eta, 4) + (1-m_avg)*(20*eta-27*eta*eta+12*pow(eta, 3)-2*pow(eta, 4))/pow((1-eta)*(2-eta), 2.0));
    double C2 = -1.*C1*C1*(m_avg*(-4*eta*eta+20*eta+8)/pow(1-eta, 5) + (1-m_avg)*(2*pow(eta, 3)+12*eta*eta-48*eta+40)/pow((1-eta)*(2-eta), 3.0));

    summ = 0.0;
    for (int i = 0; i < ncomp; i++) {
       summ += mole_fractions[i]*(components[i].getM()-1)*log(ghs[i*ncomp+i]);
    }

    double ares_hc = m_avg*ares_hs - summ;
    double ares_disp = -2*PI*den*I1*m2es3 - PI*den*m_avg*C1*I2*m2e2s3;

    summ = 0.0;
    for (int i = 0; i < ncomp; i++) {
       summ += mole_fractions[i]*(components[i].getM()-1)/ghs[i*ncomp+i]*denghs[i*ncomp+i];
    }

    double Zhc = m_avg*Zhs - summ;
    double Zdisp = -2*PI*den*detI1_det*m2es3 - PI*den*m_avg*(C1*detI2_det + C2*eta*I2)*m2e2s3;

    vector<double> dghsii_dx(ncomp*ncomp, 0);
    vector<double> dahs_dx(ncomp, 0);
    vector<double> dzeta_dx(4, 0);
    idx = -1;
    for (int i = 0; i < ncomp; i++) {
       for (int l = 0; l < 4; l++) {
           dzeta_dx[l] = PI/6.*den*components[i].getM()*pow(d[i],l);
       }
       for (int j = 0; j < ncomp; j++) {
           idx += 1;
           dghsii_dx[idx] = dzeta_dx[3]/(1-zeta[3])/(1-zeta[3]) + (d[j]*d[j]/(d[j]+d[j]))*
                   (3*dzeta_dx[2]/(1-zeta[3])/(1-zeta[3]) + 6*zeta[2]*dzeta_dx[3]/pow(1-zeta[3],3))
                   + pow(d[j]*d[j]/(d[j]+d[j]),2)*(4*zeta[2]*dzeta_dx[2]/pow(1-zeta[3],3)
                   + 6*zeta[2]*zeta[2]*dzeta_dx[3]/pow(1-zeta[3],4));
       }
       dahs_dx[i] = -dzeta_dx[0]/zeta[0]*ares_hs + 1/zeta[0]*(3*(dzeta_dx[1]*zeta[2]
               + zeta[1]*dzeta_dx[2])/(1-zeta[3]) + 3*zeta[1]*zeta[2]*dzeta_dx[3]
               /(1-zeta[3])/(1-zeta[3]) + 3*zeta[2]*zeta[2]*dzeta_dx[2]/zeta[3]/(1-zeta[3])/(1-zeta[3])
               + pow(zeta[2],3)*dzeta_dx[3]*(3*zeta[3]-1)/zeta[3]/zeta[3]/pow(1-zeta[3],3)
               + log(1-zeta[3])*((3*zeta[2]*zeta[2]*dzeta_dx[2]*zeta[3] -
               2*pow(zeta[2],3)*dzeta_dx[3])/pow(zeta[3],3) - dzeta_dx[0]) +
               (zeta[0]-pow(zeta[2],3)/zeta[3]/zeta[3])*dzeta_dx[3]/(1-zeta[3]));
    }

    vector<double> dadisp_dx(ncomp, 0);
    vector<double> dahc_dx(ncomp, 0);
    double dzeta3_dx, daa_dx, db_dx, dI1_dx, dI2_dx, dm2es3_dx, dm2e2s3_dx, dC1_dx;
    for (int i = 0; i < ncomp; i++) {
       dzeta3_dx = PI/6.*den*components[i].getM()*pow(d[i],3);
       dI1_dx = 0.0;
       dI2_dx = 0.0;
       dm2es3_dx = 0.0;
       dm2e2s3_dx = 0.0;
       for (int l = 0; l < 7; l++) {
           daa_dx = components[i].getM()/m_avg/m_avg*a1[l] + components[i].getM()/m_avg/m_avg*(3-4/m_avg)*a2[l];
           db_dx = components[i].getM()/m_avg/m_avg*b1[l] + components[i].getM()/m_avg/m_avg*(3-4/m_avg)*b2[l];
           dI1_dx += a[l]*l*dzeta3_dx*pow(eta,l-1) + daa_dx*pow(eta,l);
           dI2_dx += b[l]*l*dzeta3_dx*pow(eta,l-1) + db_dx*pow(eta,l);
       }
       for (int j = 0; j < ncomp; j++) {
           dm2es3_dx += mole_fractions[j]*components[j].getM()*(e_ij[i*ncomp+j]/_T)*pow(s_ij[i*ncomp+j],3);
           dm2e2s3_dx += mole_fractions[j]*components[j].getM()*pow(e_ij[i*ncomp+j]/_T,2)*pow(s_ij[i*ncomp+j],3);
           dahc_dx[i] += mole_fractions[j]*(components[j].getM()-1)/ghs[j*ncomp+j]*dghsii_dx[i*ncomp+j];
       }
       dm2es3_dx = dm2es3_dx*2*components[i].getM();
       dm2e2s3_dx = dm2e2s3_dx*2*components[i].getM();
       dahc_dx[i] = components[i].getM()*ares_hs + m_avg*dahs_dx[i] - dahc_dx[i] - (components[i].getM()-1)*log(ghs[i*ncomp+i]);
       dC1_dx = C2*dzeta3_dx - C1*C1*(components[i].getM()*(8*eta-2*eta*eta)/pow(1-eta,4) -
           components[i].getM()*(20*eta-27*eta*eta+12*pow(eta,3)-2*pow(eta,4))/pow((1-eta)*(2-eta),2));

       dadisp_dx[i] = -2*PI*den*(dI1_dx*m2es3 + I1*dm2es3_dx) - PI*den
           *((components[i].getM()*C1*I2 + m_avg*dC1_dx*I2 + m_avg*C1*dI2_dx)*m2e2s3
           + m_avg*C1*I2*dm2e2s3_dx);
    }

    vector<double> mu_hc(ncomp, 0);
    vector<double> mu_disp(ncomp, 0);
    for (int i = 0; i < ncomp; i++) {
       for (int j = 0; j < ncomp; j++) {
           mu_hc[i] += mole_fractions[j]*dahc_dx[j];
           mu_disp[i] += mole_fractions[j]*dadisp_dx[j];
       }
       mu_hc[i] = ares_hc + Zhc + dahc_dx[i] - mu_hc[i];
       mu_disp[i] = ares_disp + Zdisp + dadisp_dx[i] - mu_disp[i];
    }

    // Dipole term (Gross and Vrabec term) --------------------------------------
    vector<double> mu_polar(ncomp, 0);
    if (polar_term) {
       double A2 = 0.;
       double A3 = 0.;
       double dA2_det = 0.;
       double dA3_det = 0.;
       vector<double> dA2_dx(ncomp, 0);
       vector<double> dA3_dx(ncomp, 0);

       static double a0dip[5] = { 0.3043504, -0.1358588, 1.4493329, 0.3556977, -2.0653308 };
       static double a1dip[5] = { 0.9534641, -1.8396383, 2.0131180, -7.3724958, 8.2374135 };
       static double a2dip[5] = { -1.1610080, 4.5258607, 0.9751222, -12.281038, 5.9397575 };
       static double b0dip[5] = { 0.2187939, -1.1896431, 1.1626889, 0, 0 };
       static double b1dip[5] = { -0.5873164, 1.2489132, -0.5085280, 0, 0 };
       static double b2dip[5] = { 3.4869576, -14.915974, 15.372022, 0, 0 };
       static double c0dip[5] = { -0.0646774, 0.1975882, -0.8087562, 0.6902849, 0 };
       static double c1dip[5] = { -0.9520876, 2.9924258, -2.3802636, -0.2701261, 0 };
       static double c2dip[5] = { -0.6260979, 1.2924686, 1.6542783, -3.4396744, 0 };

       const static double conv = 7242.702976750923; // conversion factor, see the note below Table 2 in Gross and Vrabec 2006

       vector<double> dipmSQ (ncomp, 0);
       for (int i = 0; i < ncomp; i++) {
           dipmSQ[i] = pow(components[i].getDipm(), 2.)/(components[i].getM()*components[i].getU()*pow(components[i].getSigma(),3.))*conv;
       }

       vector<double> adip (5, 0);
       vector<double> bdip (5, 0);
       vector<double> cdip (5, 0);
       double J2, dJ2_det, J3, dJ3_det;
       double m_ij;
       double m_ijk;
       for (int i = 0; i < ncomp; i++) {
           for (int j = 0; j < ncomp; j++) {
               m_ij = sqrt(components[i].getM()*components[j].getM());
               if (m_ij > 2) {
                   m_ij = 2;
               }
               J2 = 0.;
               dJ2_det = 0.;
               for (int l = 0; l < 5; l++) {
                   adip[l] = a0dip[l] + (m_ij-1)/m_ij*a1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*a2dip[l];
                   bdip[l] = b0dip[l] + (m_ij-1)/m_ij*b1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*b2dip[l];
                   J2 += (adip[l] + bdip[l]*e_ij[j*ncomp+j]/_T)*pow(eta, l); // j*ncomp+j needs to be used for e_ij because it is formatted as a 1D vector
                   dJ2_det += (adip[l] + bdip[l]*e_ij[j*ncomp+j]/_T)*l*pow(eta, l-1);
               }
               A2 += mole_fractions[i]*mole_fractions[j]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)/
                   pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*J2;
               dA2_det += mole_fractions[i]*mole_fractions[j]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*pow(s_ij[i*ncomp+i],3)*
                   pow(s_ij[j*ncomp+j],3)/pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*dJ2_det;
               if (i == j) {
                   dA2_dx[i] += e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)
                       /pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*
                       (mole_fractions[i]*mole_fractions[j]*dJ2_det*PI/6.*den*components[i].getM()*pow(d[i],3) + 2*mole_fractions[j]*J2);
               }
               else {
                   dA2_dx[i] += e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)
                       /pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*
                       (mole_fractions[i]*mole_fractions[j]*dJ2_det*PI/6.*den*components[i].getM()*pow(d[i],3) + mole_fractions[j]*J2);
               }

               for (int k = 0; k < ncomp; k++) {
                   m_ijk = pow((components[i].getM()*components[j].getM()*components[k].getM()),1/3.);
                   if (m_ijk > 2) {
                       m_ijk = 2;
                   }
                   J3 = 0.;
                   dJ3_det = 0.;
                   for (int l = 0; l < 5; l++) {
                       cdip[l] = c0dip[l] + (m_ijk-1)/m_ijk*c1dip[l] + (m_ijk-1)/m_ijk*(m_ijk-2)/m_ijk*c2dip[l];
                       J3 += cdip[l]*pow(eta, l);
                       dJ3_det += cdip[l]*l*pow(eta, (l-1));
                   }
                   A3 += mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*
                       pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
                       s_ij[j*ncomp+k]*components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*
                       dipmSQ[j]*dipmSQ[k]*J3;
                   dA3_det += mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*
                       pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
                       s_ij[j*ncomp+k]*components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*
                       dipmSQ[j]*dipmSQ[k]*dJ3_det;
                   if ((i == j) && (i == k)) {
                       dA3_dx[i] += e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*pow(s_ij[i*ncomp+i],3)
                           *pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/s_ij[j*ncomp+k]
                           *components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*dipmSQ[j]
                           *dipmSQ[k]*(mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*dJ3_det*PI/6.*den*components[i].getM()*pow(d[i],3)
                           + 3*mole_fractions[j]*mole_fractions[k]*J3);
                   }
                   else if ((i == j) || (i == k)) {
                       dA3_dx[i] += e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*pow(s_ij[i*ncomp+i],3)
                           *pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/s_ij[j*ncomp+k]
                           *components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*dipmSQ[j]
                           *dipmSQ[k]*(mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*dJ3_det*PI/6.*den*components[i].getM()*pow(d[i],3)
                           + 2*mole_fractions[j]*mole_fractions[k]*J3);
                   }
                   else {
                       dA3_dx[i] += e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*pow(s_ij[i*ncomp+i],3)
                           *pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/s_ij[j*ncomp+k]
                           *components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*dipmSQ[j]
                           *dipmSQ[k]*(mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*dJ3_det*PI/6.*den*components[i].getM()*pow(d[i],3)
                           + mole_fractions[j]*mole_fractions[k]*J3);
                   }
               }
           }
       }

       A2 = -PI*den*A2;
       A3 = -4/3.*PI*PI*den*den*A3;
       dA2_det = -PI*den*dA2_det;
       dA3_det = -4/3.*PI*PI*den*den*dA3_det;
       for (int i = 0; i < ncomp; i++) {
           dA2_dx[i] = -PI*den*dA2_dx[i];
           dA3_dx[i] = -4/3.*PI*PI*den*den*dA3_dx[i];
       }

       vector<double> dapolar_dx(ncomp);
       for (int i = 0; i < ncomp; i++) {
           dapolar_dx[i] = (dA2_dx[i]*(1-A3/A2) + (dA3_dx[i]*A2 - A3*dA2_dx[i])/A2)/pow(1-A3/A2,2);
       }

       double ares_polar = A2/(1-A3/A2);
       double Zpolar = eta*((dA2_det*(1-A3/A2)+(dA3_det*A2-A3*dA2_det)/A2)/(1-A3/A2)/(1-A3/A2));
       for (int i = 0; i < ncomp; i++) {
           for (int j = 0; j < ncomp; j++) {
               mu_polar[i] += mole_fractions[j]*dapolar_dx[j];
           }
           mu_polar[i] = ares_polar + Zpolar + dapolar_dx[i] - mu_polar[i];
       }
    }

    // Association term -------------------------------------------------------
    // only the 2B association type is currently implemented
    vector<double> mu_assoc(ncomp, 0);
    if (assoc_term) {
       int a_sites = 2;
       int ncA = 0; // number of associating compounds
       vector<int> iA; // indices of associating compounds
       for (int i = 0; i < ncomp; i++) {
           if (components[i].getVolA() != 0) {
               iA.push_back(i);
               ncA += 1;
           }
       }

       vector<double> XA (ncA*a_sites, 0);
       vector<double> eABij (ncA*ncA, 0);
       vector<double> volABij (ncA*ncA, 0);
       vector<double> delta_ij (ncA*ncA, 0);
       vector<double> ddelta_dd (ncA*ncA*ncomp, 0);

       // these indices are necessary because we are only using 1D vectors
       int idxa = -1; // index over only associating compounds
       int idxi = 0; // index for the ii-th compound
       int idxj = 0; // index for the jj-th compound
       int idx_ddelta = -1; // index for ddelta_dd vector
       double dghsd_dd;
       for (int i = 0; i < ncA; i++) {
           idxi = iA[i]*ncomp+iA[i];
           for (int j = 0; j < ncA; j++) {
               idxa += 1;
               idxj = iA[j]*ncomp+iA[j];
               eABij[idxa] = (components[iA[i]].getUAB()+components[iA[j]].getUAB())/2.;
               volABij[idxa] = sqrt(components[iA[i]].getVolA()*components[iA[j]].getVolA())*pow(sqrt(s_ij[idxi]*
                   s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3);
               delta_ij[idxa] = ghs[iA[i]*ncomp+iA[j]]*(exp(eABij[idxa]/_T)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij[idxa];
               for (int k = 0; k < ncomp; k++) {
                   idx_ddelta += 1;
                   dghsd_dd = PI/6.*components[k].getM()*(pow(d[k], 3)/(1-zeta[3])/(1-zeta[3]) + 3*d[iA[i]]*d[iA[j]]/
                       (d[iA[i]]+d[iA[j]])*(d[k]*d[k]/(1-zeta[3])/(1-zeta[3])+2*pow(d[k], 3)*
                       zeta[2]/pow(1-zeta[3], 3)) + 2*pow((d[iA[i]]*d[iA[j]]/(d[iA[i]]+d[iA[j]])), 2)*
                       (2*d[k]*d[k]*zeta[2]/pow(1-zeta[3], 3)+3*(pow(d[k], 3)*zeta[2]*zeta[2]
                       /pow(1-zeta[3], 4))));
                   ddelta_dd[idx_ddelta] = dghsd_dd*(exp(eABij[idxa]/_T)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij[idxa];
               }
           }
           XA[i*2] = (-1 + sqrt(1+8*den*delta_ij[i*ncA+i]))/(4*den*delta_ij[i*ncA+i]);
           if (!ValidNumber(XA[i*2])) {
               XA[i*2] = 0.02;
           }
           XA[i*2+1] = XA[i*2];
       }

       vector<double> x_assoc(ncA); // mole fractions of only the associating compounds
       for (int i = 0; i < ncA; i++) {
           x_assoc[i] = mole_fractions[iA[i]];
       }

       int ctr = 0;
       double dif = 1000.;
       vector<double> XA_old = XA;
       while ((ctr < 500) && (dif > 1e-9)) {
           ctr += 1;
           XA = XA_find(XA, ncA, delta_ij, den, x_assoc);
           dif = 0.;
           for (int i = 0; i < ncA*2; i++) {
               dif += abs(XA[i] - XA_old[i]);
           }
           XA_old = XA;
       }

       vector<double> dXA_dd(ncA*a_sites*ncomp, 0);
       dXA_dd = dXA_find(ncA, ncomp, iA, delta_ij, den, XA, ddelta_dd, x_assoc, a_sites);

       for (int i = 0; i < ncomp; i++) {
           for (int j = 0; j < ncA; j++) {
               for (int k = 0; k < a_sites; k++) {
                   mu_assoc[i] += mole_fractions[iA[j]]*den*dXA_dd[i*(ncA*a_sites)+j*a_sites+k]*(1/XA[j*a_sites+k]-0.5);
               }
           }
       }

       for (int i = 0; i < ncA; i++) {
           for (int l = 0; l < a_sites; l++) {
               mu_assoc[iA[i]] += log(XA[i*a_sites+l])-0.5*XA[i*a_sites+l];
           }
           mu_assoc[iA[i]] += 0.5*a_sites;
       }
    }

    // Ion term ---------------------------------------------------------------
    vector<double> mu_ion(ncomp, 0);
    if (ion_term) {
        vector<double> q(ncomp);
        for (int i = 0; i < ncomp; i++) {
            q[i] = components[i].getZ()*E_CHRG;
        }

       summ = 0.;
       for (int i = 0; i < ncomp; i++) {
           summ += components[i].getZ()*components[i].getZ()*mole_fractions[i];
       }
       double kappa = sqrt(den*E_CHRG*E_CHRG/kb/_T/(dielc*perm_vac)*summ); // the inverse Debye screening length. Equation 4 in Held et al. 2008.

       if (kappa != 0) {
           vector<double> chi(ncomp);
           vector<double> sigma_k(ncomp);
           double summ1 = 0.;
           double summ2 = 0.;
           for (int i = 0; i < ncomp; i++) {
               chi[i] = 3/pow(kappa*components[i].getSigma(), 3)*(1.5 + log(1+kappa*components[i].getSigma()) - 2*(1+kappa*components[i].getSigma()) +
                   0.5*pow(1+kappa*components[i].getSigma(), 2));
               sigma_k[i] = -2*chi[i]+3/(1+kappa*components[i].getSigma());
               summ1 += q[i]*q[i]*mole_fractions[i]*sigma_k[i];
               summ2 += mole_fractions[i]*q[i]*q[i];
           }

           for (int i = 0; i < ncomp; i++) {
               mu_ion[i] = -q[i]*q[i]*kappa/24./PI/kb/_T/(dielc*perm_vac)*
                   (2*chi[i] + summ1/summ2);
           }
       }
    }

    CoolPropDbl Z = calc_compressibility_factor();

    vector<double> mu(ncomp, 0);
    vector<CoolPropDbl> fugcoef(ncomp, 0);
    for (int i = 0; i < ncomp; i++) {
       mu[i] = mu_hc[i] + mu_disp[i] + mu_polar[i] + mu_assoc[i] + mu_ion[i];
       fugcoef[i] = exp(mu[i] - log(Z)); // the fugacity coefficients
    }

    return fugcoef;
    }

CoolPropDbl PCSAFTBackend::calc_gibbsmolar_residual(void) {
    CoolPropDbl ares = calc_alphar();
    CoolPropDbl Z = calc_compressibility_factor();

    CoolPropDbl gres = (ares + (Z - 1) - log(Z))*kb*N_AV*_T; // Equation A.50 from Gross and Sadowski 2001
    return gres;
}

CoolPropDbl PCSAFTBackend::calc_compressibility_factor(void){
    int ncomp = N; // number of components
    vector<double> d(ncomp);
    for (int i = 0; i < ncomp; i++) {
        d[i] = components[i].getSigma()*(1-0.12*exp(-3*components[i].getU()/_T));
    }
    if (ion_term) {
        for (int i = 0; i < ncomp; i++) {
            if (components[i].getZ() != 0) {
                d[i] = components[i].getSigma()*(1-0.12); // for ions the diameter is assumed to be temperature independent (see Held et al. 2014)
            }
        }
    }

    double den = _rhomolar*N_AV/1.0e30;

    vector<double> zeta(4, 0);
    double summ;
    for (int i = 0; i < 4; i++) {
        summ = 0;
        for (int j = 0; j < ncomp; j++) {
            summ += mole_fractions[j]*components[j].getM()*pow(d[j], i);
        }
        zeta[i] = PI/6*den*summ;
    }

    double eta = zeta[3];
    double m_avg = 0;
    for (int i = 0; i < ncomp; i++) {
        m_avg += mole_fractions[i]*components[i].getM();
    }

    vector<double> ghs (ncomp, 0);
    vector<double> denghs (ncomp, 0);
    vector<double> e_ij (ncomp*ncomp, 0);
    vector<double> s_ij (ncomp*ncomp, 0);
    double m2es3 = 0.;
    double m2e2s3 = 0.;
    int idx = -1;
    for (int i = 0; i < ncomp; i++) {
        for (int j = 0; j < ncomp; j++) {
            idx += 1;
            s_ij[idx] = (components[i].getSigma() + components[j].getSigma())/2.;
            if (ion_term) {
                if (components[i].getZ()*components[j].getZ() <= 0) { // for two cations or two anions e_ij is kept at zero to avoid dispersion between like ions (see Held et al. 2014)
                    if (k_ij.empty()) {
                        e_ij[idx] = sqrt(components[i].getU()*components[j].getU());
                    }
                    else {
                        e_ij[idx] = sqrt(components[i].getU()*components[j].getU())*(1 - (k_ij[idx] + k_ijT[idx] * _T));
                    }
                }
            } else {
                if (k_ij.empty()) {
                    e_ij[idx] = sqrt(components[i].getU()*components[j].getU());
                }
                else {
                    e_ij[idx] = sqrt(components[i].getU()*components[j].getU())*(1 - (k_ij[idx] + k_ijT[idx] * _T));
                }
            }
            m2es3 = m2es3 + mole_fractions[i]*mole_fractions[j]*components[i].getM()*components[j].getM()*e_ij[idx]/_T*pow(s_ij[idx], 3);
            m2e2s3 = m2e2s3 + mole_fractions[i]*mole_fractions[j]*components[i].getM()*components[j].getM()*pow(e_ij[idx]/_T,2)*pow(s_ij[idx], 3);
        }
        ghs[i] = 1/(1-zeta[3]) + (d[i]*d[i]/(d[i]+d[i]))*3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
            pow(d[i]*d[i]/(d[i]+d[i]), 2)*2*zeta[2]*zeta[2]/pow(1-zeta[3], 3);
        denghs[i] = zeta[3]/(1-zeta[3])/(1-zeta[3]) +
            (d[i]*d[i]/(d[i]+d[i]))*(3*zeta[2]/(1-zeta[3])/(1-zeta[3]) +
            6*zeta[2]*zeta[3]/pow(1-zeta[3], 3)) +
            pow(d[i]*d[i]/(d[i]+d[i]), 2)*(4*zeta[2]*zeta[2]/pow(1-zeta[3], 3) +
            6*zeta[2]*zeta[2]*zeta[3]/pow(1-zeta[3], 4));
    }

    double Zhs = zeta[3]/(1-zeta[3]) + 3.*zeta[1]*zeta[2]/zeta[0]/(1.-zeta[3])/(1.-zeta[3]) +
        (3.*pow(zeta[2], 3.) - zeta[3]*pow(zeta[2], 3.))/zeta[0]/pow(1.-zeta[3], 3.);

    static double a0[7] = { 0.910563145, 0.636128145, 2.686134789, -26.54736249, 97.75920878, -159.5915409, 91.29777408 };
    static double a1[7] = { -0.308401692, 0.186053116, -2.503004726, 21.41979363, -65.25588533, 83.31868048, -33.74692293 };
    static double a2[7] = { -0.090614835, 0.452784281, 0.596270073, -1.724182913, -4.130211253, 13.77663187, -8.672847037 };
    static double b0[7] = { 0.724094694, 2.238279186, -4.002584949, -21.00357682, 26.85564136, 206.5513384, -355.6023561 };
    static double b1[7] = { -0.575549808, 0.699509552, 3.892567339, -17.21547165, 192.6722645, -161.8264617, -165.2076935 };
    static double b2[7] = { 0.097688312, -0.255757498, -9.155856153, 20.64207597, -38.80443005, 93.62677408, -29.66690559 };

    vector<double> a(7, 0);
    vector<double> b(7, 0);
    for (int i = 0; i < 7; i++) {
        a[i] = a0[i] + (m_avg-1.)/m_avg*a1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*a2[i];
        b[i] = b0[i] + (m_avg-1.)/m_avg*b1[i] + (m_avg-1.)/m_avg*(m_avg-2.)/m_avg*b2[i];
    }

    double detI1_det = 0.0;
    double detI2_det = 0.0;
    double I2 = 0.0;
    for (int i = 0; i < 7; i++) {
        detI1_det += a[i]*(i+1)*pow(eta, i);
        detI2_det += b[i]*(i+1)*pow(eta, i);
        I2 += b[i]*pow(eta, i);
    }
    double C1 = 1./(1. + m_avg*(8*eta-2*eta*eta)/pow(1-eta, 4) + (1-m_avg)*(20*eta-27*eta*eta+12*pow(eta, 3)-2*pow(eta, 4))/pow((1-eta)*(2-eta), 2.0));
    double C2 = -1.*C1*C1*(m_avg*(-4*eta*eta+20*eta+8)/pow(1-eta, 5) + (1-m_avg)*(2*pow(eta, 3)+12*eta*eta-48*eta+40)/pow((1-eta)*(2-eta), 3.0));

    summ = 0.0;
    for (int i = 0; i < ncomp; i++) {
        summ += mole_fractions[i]*(components[i].getM()-1)/ghs[i]*denghs[i];
    }

    double Zid = 1.0;
    double Zhc = m_avg*Zhs - summ;
    double Zdisp = -2*PI*den*detI1_det*m2es3 - PI*den*m_avg*(C1*detI2_det + C2*eta*I2)*m2e2s3;

    // Dipole term (Gross and Vrabec term) --------------------------------------
    double Zpolar = 0;
    if (polar_term) {
        double A2 = 0.;
        double A3 = 0.;
        double dA2_det = 0.;
        double dA3_det = 0.;
        vector<double> adip (5, 0);
        vector<double> bdip (5, 0);
        vector<double> cdip (5, 0);
        vector<double> dipmSQ (ncomp, 0);
        double J2, dJ2_det, J3, dJ3_det;

        static double a0dip[5] = { 0.3043504, -0.1358588, 1.4493329, 0.3556977, -2.0653308 };
        static double a1dip[5] = { 0.9534641, -1.8396383, 2.0131180, -7.3724958, 8.2374135 };
        static double a2dip[5] = { -1.1610080, 4.5258607, 0.9751222, -12.281038, 5.9397575 };
        static double b0dip[5] = { 0.2187939, -1.1896431, 1.1626889, 0, 0 };
        static double b1dip[5] = { -0.5873164, 1.2489132, -0.5085280, 0, 0 };
        static double b2dip[5] = { 3.4869576, -14.915974, 15.372022, 0, 0 };
        static double c0dip[5] = { -0.0646774, 0.1975882, -0.8087562, 0.6902849, 0 };
        static double c1dip[5] = { -0.9520876, 2.9924258, -2.3802636, -0.2701261, 0 };
        static double c2dip[5] = { -0.6260979, 1.2924686, 1.6542783, -3.4396744, 0 };

        const static double conv = 7242.702976750923; // conversion factor, see the note below Table 2 in Gross and Vrabec 2006

        for (int i = 0; i < ncomp; i++) {
            dipmSQ[i] = pow(components[i].getDipm(), 2.)/(components[i].getM()*components[i].getU()*pow(components[i].getSigma(),3.))*conv;
        }

        double m_ij;
        for (int i = 0; i < ncomp; i++) {
            for (int j = 0; j < ncomp; j++) {
                m_ij = sqrt(components[i].getM()*components[j].getM());
                if (m_ij > 2) {
                    m_ij = 2;
                }
                J2 = 0.;
                dJ2_det = 0.;
                for (int l = 0; l < 5; l++) {
                    adip[l] = a0dip[l] + (m_ij-1)/m_ij*a1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*a2dip[l];
                    bdip[l] = b0dip[l] + (m_ij-1)/m_ij*b1dip[l] + (m_ij-1)/m_ij*(m_ij-2)/m_ij*b2dip[l];
                    J2 += (adip[l] + bdip[l]*e_ij[j*ncomp+j]/_T)*pow(eta, l); // j*ncomp+j needs to be used for e_ij because it is formatted as a 1D vector
                    dJ2_det += (adip[l] + bdip[l]*e_ij[j*ncomp+j]/_T)*l*pow(eta, l-1);
                }
                A2 += mole_fractions[i]*mole_fractions[j]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)/
                    pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*J2;
                dA2_det += mole_fractions[i]*mole_fractions[j]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*pow(s_ij[i*ncomp+i],3)*
                    pow(s_ij[j*ncomp+j],3)/pow(s_ij[i*ncomp+j],3)*components[i].getDipnum()*components[j].getDipnum()*dipmSQ[i]*dipmSQ[j]*dJ2_det;
            }
        }

        double m_ijk;
        for (int i = 0; i < ncomp; i++) {
            for (int j = 0; j < ncomp; j++) {
                for (int k = 0; k < ncomp; k++) {
                    m_ijk = pow((components[i].getM()*components[j].getM()*components[k].getM()),1/3.);
                    if (m_ijk > 2) {
                        m_ijk = 2;
                    }
                    J3 = 0.;
                    dJ3_det = 0.;
                    for (int l = 0; l < 5; l++) {
                        cdip[l] = c0dip[l] + (m_ijk-1)/m_ijk*c1dip[l] + (m_ijk-1)/m_ijk*(m_ijk-2)/m_ijk*c2dip[l];
                        J3 += cdip[l]*pow(eta, l);
                        dJ3_det += cdip[l]*l*pow(eta, (l-1));
                    }
                    A3 += mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*
                        pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
                        s_ij[j*ncomp+k]*components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*
                        dipmSQ[j]*dipmSQ[k]*J3;
                    dA3_det += mole_fractions[i]*mole_fractions[j]*mole_fractions[k]*e_ij[i*ncomp+i]/_T*e_ij[j*ncomp+j]/_T*e_ij[k*ncomp+k]/_T*
                        pow(s_ij[i*ncomp+i],3)*pow(s_ij[j*ncomp+j],3)*pow(s_ij[k*ncomp+k],3)/s_ij[i*ncomp+j]/s_ij[i*ncomp+k]/
                        s_ij[j*ncomp+k]*components[i].getDipnum()*components[j].getDipnum()*components[k].getDipnum()*dipmSQ[i]*
                        dipmSQ[j]*dipmSQ[k]*dJ3_det;
                }
            }
        }

        A2 = -PI*den*A2;
        A3 = -4/3.*PI*PI*den*den*A3;
        dA2_det = -PI*den*dA2_det;
        dA3_det = -4/3.*PI*PI*den*den*dA3_det;

        Zpolar = eta*((dA2_det*(1-A3/A2)+(dA3_det*A2-A3*dA2_det)/A2)/(1-A3/A2)/(1-A3/A2));
    }

    // Association term -------------------------------------------------------
    // only the 2B association type is currently implemented
    double Zassoc = 0;
    if (assoc_term) {
        int a_sites = 2;
        int ncA = 0; // number of associating compounds
        vector<int> iA; // indices of associating compounds
        for (int i = 0; i < ncomp; i++) {
            if (components[i].getVolA() != 0) {
                iA.push_back(i);
                ncA += 1;
            }
        }

        vector<double> XA (ncA*a_sites, 0);
        vector<double> eABij (ncA*ncA, 0);
        vector<double> volABij (ncA*ncA, 0);
        vector<double> delta_ij (ncA*ncA, 0);
        vector<double> ddelta_dd (ncA*ncA*ncomp, 0);

        // these indices are necessary because we are only using 1D vectors
        int idxa = -1; // index over only associating compounds
        int idxi = 0; // index for the ii-th compound
        int idxj = 0; // index for the jj-th compound
        int idx_ddelta = -1; // index for ddelta_dd vector
        double dghsd_dd;
        for (int i = 0; i < ncA; i++) {
            idxi = iA[i]*ncomp+iA[i];
            for (int j = 0; j < ncA; j++) {
                idxa += 1;
                idxj = iA[j]*ncomp+iA[j];
                eABij[idxa] = (components[iA[i]].getUAB()+components[iA[j]].getUAB())/2.;
                volABij[idxa] = sqrt(components[iA[i]].getVolA()*components[iA[j]].getVolA())*pow(sqrt(s_ij[idxi]*
                    s_ij[idxj])/(0.5*(s_ij[idxi]+s_ij[idxj])), 3);
                delta_ij[idxa] = ghs[iA[j]]*(exp(eABij[idxa]/_T)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij[idxa];
                for (int k = 0; k < ncomp; k++) {
                    idx_ddelta += 1;
                    dghsd_dd = PI/6.*components[k].getM()*(pow(d[k], 3)/(1-zeta[3])/(1-zeta[3]) + 3*d[iA[i]]*d[iA[j]]/
                        (d[iA[i]]+d[iA[j]])*(d[k]*d[k]/(1-zeta[3])/(1-zeta[3])+2*pow(d[k], 3)*
                        zeta[2]/pow(1-zeta[3], 3)) + 2*pow((d[iA[i]]*d[iA[j]]/(d[iA[i]]+d[iA[j]])), 2)*
                        (2*d[k]*d[k]*zeta[2]/pow(1-zeta[3], 3)+3*(pow(d[k], 3)*zeta[2]*zeta[2]
                        /pow(1-zeta[3], 4))));
                    ddelta_dd[idx_ddelta] = dghsd_dd*(exp(eABij[idxa]/_T)-1)*pow(s_ij[iA[i]*ncomp+iA[j]], 3)*volABij[idxa];
                }
            }
            XA[i*2] = (-1 + sqrt(1+8*den*delta_ij[i*ncA+i]))/(4*den*delta_ij[i*ncA+i]);
            XA[i*2+1] = XA[i*2];
        }

        vector<double> x_assoc(ncA); // mole fractions of only the associating compounds
        for (int i = 0; i < ncA; i++) {
            x_assoc[i] = mole_fractions[iA[i]];
        }

        int ctr = 0;
        double dif = 1000.;
        vector<double> XA_old = XA;
        while ((ctr < 500) && (dif > 1e-9)) {
            ctr += 1;
            XA = XA_find(XA, ncA, delta_ij, den, x_assoc);
            dif = 0.;
            for (int i = 0; i < ncA*2; i++) {
                dif += abs(XA[i] - XA_old[i]);
            }
            XA_old = XA;
        }

        vector<double> dXA_dd(a_sites*ncA*ncomp, 0);
        dXA_dd = dXA_find(ncA, ncomp, iA, delta_ij, den, XA, ddelta_dd, x_assoc, a_sites);

        summ = 0.;
        for (int i = 0; i < ncomp; i++) {
            for (int j = 0; j < ncA; j++) {
                for (int k = 0; k < a_sites; k++) {
                    summ += mole_fractions[i]*den*mole_fractions[iA[j]]*(1/XA[j*a_sites+k]-0.5)*dXA_dd[i*(ncA*a_sites)+j*(a_sites)+k];
                }
            }
        }

        Zassoc = summ;
    }

    // Ion term ---------------------------------------------------------------
    double Zion = 0;
    if (ion_term) {
        vector<double> q(ncomp);
        for (int i = 0; i < ncomp; i++) {
            q[i] = components[i].getZ()*E_CHRG;
        }

        summ = 0.;
        for (int i = 0; i < ncomp; i++) {
            summ += pow(components[i].getZ(),2.)*mole_fractions[i];
        }

        double kappa = sqrt(den*E_CHRG*E_CHRG/kb/_T/(dielc*perm_vac)*summ); // the inverse Debye screening length. Equation 4 in Held et al. 2008.

        if (kappa != 0) {
            double chi, sigma_k;
            summ = 0.;
            for (int i = 0; i < ncomp; i++) {
                chi = 3/pow(kappa*components[i].getSigma(), 3)*(1.5 +
                    log(1+kappa*components[i].getSigma()) - 2*(1+kappa*components[i].getSigma()) +
                    0.5*pow(1+kappa*components[i].getSigma(), 2));
                sigma_k = -2*chi+3/(1+kappa*components[i].getSigma());
                summ += q[i]*q[i]*mole_fractions[i]*sigma_k;
            }
            Zion = -1*kappa/24./PI/kb/_T/(dielc*perm_vac)*summ;
        }
    }

    double Z = Zid + Zhc + Zdisp + Zpolar + Zassoc + Zion;
    return Z;
}

void PCSAFTBackend::post_update(bool optional_checks){
    // Check the values that must always be set
    // if (_p < 0){ throw ValueError("p is less than zero");}
    if (!ValidNumber(_p)){
        throw ValueError("p is not a valid number");}
    if (_T < 0){ throw ValueError("T is less than zero");}
    if (!ValidNumber(_T)){ throw ValueError("T is not a valid number");}
    if (_rhomolar < 0){ throw ValueError("rhomolar is less than zero");}
    if (!ValidNumber(_rhomolar)){ throw ValueError("rhomolar is not a valid number");}

    if (optional_checks){
        if (!ValidNumber(_Q)){ throw ValueError("Q is not a valid number");}
        if (_phase == iphase_unknown){
                throw ValueError("_phase is unknown");
        }
    }
}

void PCSAFTBackend::update(CoolProp::input_pairs input_pair, double value1, double value2){
    if (get_debug_level() > 10){std::cout << format("%s (%d): update called with (%d: (%s), %g, %g)",__FILE__,__LINE__, input_pair, get_input_pair_short_desc(input_pair).c_str(), value1, value2) << std::endl;}

    // Converting input to CoolPropDbl
    CoolPropDbl ld_value1 = value1, ld_value2 = value2;
    value1 = ld_value1; value2 = ld_value2;

    // Clear the state
    clear();

    if (is_pure_or_pseudopure == false && mole_fractions.size() == 0) {
        throw ValueError("Mole fractions must be set");
    }

    if (SatL->mole_fractions.empty()) {
        SatL->set_mole_fractions(mole_fractions);
    }
    if (SatV->mole_fractions.empty()) {
        SatV->set_mole_fractions(mole_fractions);
        double summ = 0;
        for (int i = 0; i < N; i++) {
            if (SatV->components[i].getZ() != 0) { // we make the assumption that ions do not appear in the vapor phase
                summ -= SatV->mole_fractions[i];
                SatV->mole_fractions[i] = 0;
            }
        }
        for (int i = 0; i < N; i++) {
            SatV->mole_fractions[i] = SatV->mole_fractions[i] / summ;
        }
    }

    // If the inputs are in mass units, convert them to molar units
    mass_to_molar_inputs(input_pair, value1, value2);

    switch(input_pair)
    {
        case PT_INPUTS:
            _p = value1; _T = value2;
            if (water_present) {
                components[water_idx].calc_water_sigma(_T);
                dielc = dielc_water(_T); // Right now only aqueous mixtures are supported. Other solvents could be modeled by replacing the dielc_water function.
            }

            if (imposed_phase_index != iphase_not_imposed) {
                // Use the imposed phase index
                _phase = imposed_phase_index;
            }
            else {
                _phase = calc_phase_internal(input_pair);
            }
            _rhomolar = solver_rho_Tp(value2/*T*/, value1/*p*/, _phase/*phase*/); break;
        case QT_INPUTS:
            _Q = value1; _T = value2;
            SatL->_Q = value1; SatV->_Q = value1;
            SatL->_T = value2; SatV->_T = value2;
            _phase = iphase_twophase;
            if ((_Q < 0) || (_Q > 1))
                throw CoolProp::OutOfRangeError("Input vapor quality [Q] must be between 0 and 1");
            if (water_present) {
                components[water_idx].calc_water_sigma(_T);
                SatL->components[water_idx].calc_water_sigma(_T);
                SatV->components[water_idx].calc_water_sigma(_T);
                dielc = dielc_water(_T); // Right now only aqueous mixtures are supported. Other solvents could be modeled by replacing the dielc_water function.
                SatL->dielc = dielc_water(_T);
                SatV->dielc = dielc_water(_T);
            }
            flash_QT(*this); break;
        case PQ_INPUTS:
            _p = value1; _Q = value2;
            SatL->_p = value1; SatV->_p = value1;
            SatL->_Q = value2; SatV->_Q = value2;
            _phase = iphase_twophase;
            if ((_Q < 0) || (_Q > 1)) {
                throw CoolProp::OutOfRangeError("Input vapor quality [Q] must be between 0 and 1");
            }
            flash_PQ(*this); break;
        case DmolarT_INPUTS:
            _rhomolar = value1; _T = value2;
            if (water_present) {
                components[water_idx].calc_water_sigma(_T);
                dielc = dielc_water(_T); // Right now only aqueous mixtures are supported. Other solvents could be modeled by replacing the dielc_water function.
            }
            _p = update_DmolarT(_rhomolar);

            if (imposed_phase_index != iphase_not_imposed) {
                // Use the imposed phase index
                _phase = imposed_phase_index;
            }
            else {
                _phase = calc_phase_internal(input_pair); // if in the two phase region, the pressure is updated by this function to equal the bubble point
            }
            break;
        case SmolarT_INPUTS:
        case DmolarP_INPUTS:
        case DmolarHmolar_INPUTS:
        case DmolarSmolar_INPUTS:
        case DmolarUmolar_INPUTS:
        case HmolarP_INPUTS:
        case PSmolar_INPUTS:
        case PUmolar_INPUTS:
        case HmolarSmolar_INPUTS:
        case QSmolar_INPUTS:
        case HmolarQ_INPUTS:
        case DmolarQ_INPUTS:
        default:
            throw ValueError(format("This pair of inputs [%s] is not yet supported", get_input_pair_short_desc(input_pair).c_str()));
    }

    // set Q, if not already set
    if (!ValidNumber(_Q)) {
        if (_phase == iphase_gas) {_Q = 1;}
        else if (_phase == iphase_liquid) {_Q = 0;}
    }

    post_update();
}

phases PCSAFTBackend::calc_phase_internal(CoolProp::input_pairs input_pair) {
    phases phase;

    double p_input, rho_input;
    double p_bub, p_dew;
    switch(input_pair)
    {
        case PT_INPUTS:
            p_input = _p; rho_input = _rhomolar;
            _Q = 0;
            SatL->_Q = _Q; SatV->_Q = _Q;
            SatL->_T = _T; SatV->_T = _T;
            try {
                flash_QT(*this);
            }
            catch (const SolutionError& ex) {
                phase = iphase_supercritical;
                break;
            }
            p_bub = _p;
            _p = p_input; _rhomolar = rho_input;
            if (_p > p_bub) {
                phase = iphase_liquid;
            }
            else if (_p == p_bub) {
                phase = iphase_twophase;
            }
            else {
                _Q = 1;
                SatL->_Q = _Q; SatV->_Q = _Q;
                flash_QT(*this);
                p_dew = _p;
                _p = p_input; _rhomolar = rho_input;
                if (_p < p_dew) {
                    phase = iphase_gas;
                }
                else if ((_p <= p_bub) && (_p >= p_dew)) {
                    phase = iphase_twophase;
                }
                else{
                	phase = iphase_unknown;
                }
            }
            break;
        case DmolarT_INPUTS:
            double rho_bub, rho_dew;
            p_input = _p; rho_input = _rhomolar;
            _Q = 0;
            SatL->_Q = _Q; SatV->_Q = _Q;
            SatL->_T = _T; SatV->_T = _T;
            try {
                flash_QT(*this);
            }
            catch (const SolutionError& ex) {
                phase = iphase_supercritical;
                break;
            }
            rho_bub = _rhomolar;
            p_bub = _p;
            _p = p_input; _rhomolar = rho_input;
            if (_rhomolar > rho_bub) {
                phase = iphase_liquid;
            }
            else if (_rhomolar == rho_bub) {
                phase = iphase_twophase;
                _p = p_bub;
                _Q = 1 - (_rhomolar - SatV->_rhomolar) / (SatL->_rhomolar - SatV->_rhomolar);
            }
            else {
                _Q = 1;
                SatL->_Q = _Q; SatV->_Q = _Q;
                flash_QT(*this);
                rho_dew = _rhomolar;
                _p = p_input; _rhomolar = rho_input;
                if (_rhomolar < rho_dew) {
                    phase = iphase_gas;
                }
                else if ((_rhomolar <= rho_bub) && (_rhomolar >= rho_dew)) {
                    phase = iphase_twophase;
                    _p = p_bub;
                    _Q = 1 - (_rhomolar - SatV->_rhomolar) / (SatL->_rhomolar - SatV->_rhomolar);
                }
            }
            break;
        default:
            throw ValueError(format("Phase determination for this pair of inputs [%s] is not yet supported", get_input_pair_short_desc(input_pair).c_str()));
    }

    return phase;
}

void PCSAFTBackend::flash_QT(PCSAFTBackend &PCSAFT) {
    CoolPropDbl T = PCSAFT._T;

    class SolverBubblePResid : public FuncWrapper1D {
    public:
        PCSAFTBackend &PCSAFT;
        CoolPropDbl T, p;

        SolverBubblePResid(PCSAFTBackend &PCSAFT, CoolPropDbl T)
        : PCSAFT(PCSAFT), T(T) {}
        CoolPropDbl call(CoolPropDbl p){
            double error = 0;
            if (p <= 0) {
                error = 1e20;
            }
            else {
                if (PCSAFT.is_pure_or_pseudopure) {
                    PCSAFT.SatL->_rhomolar = PCSAFT.SatL->solver_rho_Tp(T, p, iphase_liquid);
                    vector<CoolPropDbl> fugcoef_l = PCSAFT.SatL->calc_fugacity_coefficients();
                    PCSAFT.SatV->_rhomolar = PCSAFT.SatV->solver_rho_Tp(T, p, iphase_gas);
                    vector<CoolPropDbl> fugcoef_v = PCSAFT.SatV->calc_fugacity_coefficients();
                    error += 100000 * pow(fugcoef_l[0] - fugcoef_v[0], 2.);
                }
                else {
                    if (PCSAFT.N > 1) {
                        bool reset_mole_fractions = false;
                        for (int i = 0; i < PCSAFT.N; i++) {
                            if (!ValidNumber(PCSAFT.SatL->mole_fractions[i]) || !ValidNumber(PCSAFT.SatV->mole_fractions[i])) {
                                reset_mole_fractions = true;
                            }
                        }
                        if (reset_mole_fractions) {
                            PCSAFT.SatL->mole_fractions = PCSAFT.mole_fractions;
                            PCSAFT.SatV->mole_fractions = PCSAFT.mole_fractions;
                        }
                    }

                    int itr = 0;
                    double dif = 10000.;
                    vector<CoolPropDbl> fugcoef_l(PCSAFT.N), fugcoef_v(PCSAFT.N);

                    double rhol, rhov, summ;
                    vector<CoolPropDbl> xv_old(PCSAFT.N);
                    double x_ions = 0.; // overall mole fraction of ions in the system
                    for (int i = 0; i < PCSAFT.N; i++) {
                        if (PCSAFT.components[i].getZ() != 0) {
                            x_ions += PCSAFT.mole_fractions[i];
                        }
                    }
                    while ((dif>1e-9) && (itr<100)) {
                        xv_old = PCSAFT.SatV->mole_fractions;
                        PCSAFT.SatL->_rhomolar = PCSAFT.SatL->solver_rho_Tp(T, p, iphase_liquid);
                        fugcoef_l = PCSAFT.SatL->calc_fugacity_coefficients();
                        PCSAFT.SatV->_rhomolar = PCSAFT.SatV->solver_rho_Tp(T, p, iphase_gas);
                        fugcoef_v = PCSAFT.SatV->calc_fugacity_coefficients();

                        if (PCSAFT._Q > 0.5) {
                            summ = 0.;
                            for (int i = 0; i < PCSAFT.N; i++) {
                                if (PCSAFT.components[i].getZ() == 0) {
                                    PCSAFT.SatL->mole_fractions[i] = fugcoef_v[i]*PCSAFT.SatV->mole_fractions[i]/fugcoef_l[i];
                                    summ += PCSAFT.SatL->mole_fractions[i];
                                }
                            }
                            for (int i = 0; i < PCSAFT.N; i++) {
                                if (PCSAFT.components[i].getZ() == 0) {
                                    PCSAFT.SatL->mole_fractions[i] = PCSAFT.SatL->mole_fractions[i]/summ*(((1-PCSAFT._Q) - x_ions)/(1-PCSAFT._Q)); // ensures that mole fractions add up to 1
                                    PCSAFT.SatV->mole_fractions[i] = (PCSAFT.mole_fractions[i] - (1-PCSAFT._Q)*PCSAFT.SatL->mole_fractions[i])/PCSAFT._Q; // if PCSAFT->_Q is close to zero then this equation behaves poorly, and that is why we use this if statement to switch the equation around
                                }
                                else {
                                    PCSAFT.SatL->mole_fractions[i] = PCSAFT.mole_fractions[i]/(1-PCSAFT._Q);
                                    PCSAFT.SatV->mole_fractions[i] = 0.;
                                }
                            }
                        }
                        else {
                            summ = 0.;
                            for (int i = 0; i < PCSAFT.N; i++) {
                                if (PCSAFT.components[i].getZ() == 0) {
                                    PCSAFT.SatV->mole_fractions[i] = fugcoef_l[i]*PCSAFT.SatL->mole_fractions[i]/fugcoef_v[i];
                                }
                                summ += PCSAFT.SatV->mole_fractions[i];
                            }
                            for (int i = 0; i < PCSAFT.N; i++) {
                                PCSAFT.SatV->mole_fractions[i] = PCSAFT.SatV->mole_fractions[i]/summ;
                                PCSAFT.SatL->mole_fractions[i] = (PCSAFT.mole_fractions[i] - (PCSAFT._Q)*PCSAFT.SatV->mole_fractions[i])/(1-PCSAFT._Q);
                            }
                        }

                        dif = 0;
                        for (int i = 0; i < PCSAFT.N; i++) {
                            dif += abs(PCSAFT.SatV->mole_fractions[i] - xv_old[i]);
                        }
                        itr += 1;
                    }

                    for (int i = 0; i < PCSAFT.N; i++) {
                        if (PCSAFT.components[i].getZ() == 0) {
                            error += pow(PCSAFT.SatL->mole_fractions[i]*fugcoef_l[i] - PCSAFT.SatV->mole_fractions[i]*fugcoef_v[i], 2.);
                        }
                        error += pow((PCSAFT.mole_fractions[i] - PCSAFT._Q*PCSAFT.SatV->mole_fractions[i] - (1-PCSAFT._Q)*PCSAFT.SatL->mole_fractions[i]), 2.);
                    }
                }

                if (!ValidNumber(error) || (PCSAFT.SatL->_rhomolar - PCSAFT.SatV->_rhomolar) < 1e-5) {
                    error = 1e20;
                }
            }
            return error;
        };
    };

    SolverBubblePResid resid(*this, T);

    CoolPropDbl p_guess = _HUGE;
    double x_lo = _HUGE;
    double x_hi = _HUGE;
    double x_lbound = -8;
    double x_ubound = 9;

    try {
        // scanning the range of pressures to find a good initial guess
        int npts = 30;
        double err_min = 1e20;
        int ctr_increasing = 0; // keeps track of the number of steps where the error is increasing instead of decreasing
        for (int i = 0; i < npts; i++) {
            CoolPropDbl p_i = pow(10, ((x_ubound - x_lbound) / (double)npts * i + x_lbound));
            double err = resid.call(p_i);

            if (err < err_min) {
                err_min = err;
                p_guess = p_i;
                x_lo = pow(10, ((x_ubound - x_lbound) / (double)npts * (i - 1) + x_lbound));
                x_hi = pow(10, ((x_ubound - x_lbound) / (double)npts * (i + 1) + x_lbound));
                ctr_increasing = 0;
            }
            else if (err_min < 1e20) {
                ctr_increasing += 1;
            }

            if (ctr_increasing > 2) { // this is necessary because PC-SAFT often gives a second, erroneous VLE at lower temperatures. Reference: Privat R, Gani R, Jaubert JN. Are safe results obtained when the PC-SAFT equation of state is applied to ordinary pure chemicals?. Fluid Phase Equilibria. 2010 Aug 15;295(1):76-92.
                break;
            }
        }

        if (p_guess == _HUGE) {
              throw SolutionError(format("A suitable initial guess for pressure could not be found for the QT flash."));
        }
    }
    catch (const SolutionError& ex) {
        // scanning the range of pressures to find a good initial guess
        int npts = 500;
        double err_min = 1e20;
        int ctr_increasing = 0; // keeps track of the number of steps where the error is increasing instead of decreasing
        for (int i = 0; i < npts; i++) {
            CoolPropDbl p_i = pow(10, ((x_ubound - x_lbound) / (double)npts * i + x_lbound));
            double err = resid.call(p_i);

            if (err < err_min) {
                err_min = err;
                p_guess = p_i;
                x_lo = pow(10, ((x_ubound - x_lbound) / (double)npts * (i - 1) + x_lbound));
                x_hi = pow(10, ((x_ubound - x_lbound) / (double)npts * (i + 1) + x_lbound));
                ctr_increasing = 0;
            }
            else if (err_min < 1e20) {
                ctr_increasing += 1;
            }

            if (ctr_increasing > 2) { // this is necessary because PC-SAFT often gives a second, erroneous VLE at lower temperatures. Reference: Privat R, Gani R, Jaubert JN. Are safe results obtained when the PC-SAFT equation of state is applied to ordinary pure chemicals?. Fluid Phase Equilibria. 2010 Aug 15;295(1):76-92.
                break;
            }
        }

        if (p_guess == _HUGE) {
            throw SolutionError(format("A suitable initial guess for pressure could not be found for the QT flash."));
        }
    }

    CoolPropDbl p;
    try {
        p = BoundedSecant(resid, p_guess, x_lo, x_hi, 0.01*p_guess, 1e-8, 200);
    }
    catch (const SolutionError& ex) {
        p = BoundedSecant(resid, p_guess, x_lo, x_hi, 0.01*p_guess, 0.1, 200);
    }

    // Load the outputs
    PCSAFT._p = p;
    PCSAFT._rhomolar = 1/(PCSAFT._Q/PCSAFT.SatV->_rhomolar + (1 - PCSAFT._Q)/PCSAFT.SatL->_rhomolar);
    PCSAFT._phase = iphase_twophase;
}

void PCSAFTBackend::flash_PQ(PCSAFTBackend &PCSAFT){
    CoolPropDbl p = PCSAFT._p;

    class SolverTboilResid : public FuncWrapper1D {
    public:
        PCSAFTBackend &PCSAFT;
        CoolPropDbl T, p;

        SolverTboilResid(PCSAFTBackend &PCSAFT, CoolPropDbl p)
        : PCSAFT(PCSAFT), p(p) {}
        CoolPropDbl call(CoolPropDbl T){
            double error = 0;
            if (T <= 0) {
                error = 1e20;
            }
            else {
                PCSAFT.SatL->_T = T; // _T must be updated because the density calculation depends on it
                PCSAFT.SatV->_T = T;

                if (PCSAFT.water_present) {
                    try {
                        PCSAFT.components[PCSAFT.water_idx].calc_water_sigma(T);
                        PCSAFT.SatL->components[PCSAFT.water_idx].calc_water_sigma(T);
                        PCSAFT.SatV->components[PCSAFT.water_idx].calc_water_sigma(T);
                        PCSAFT.dielc = PCSAFT.dielc_water(T); // Right now only aqueous mixtures are supported. Other solvents could be modeled by replacing the dielc_water function.
                        PCSAFT.SatL->dielc = PCSAFT.dielc_water(T);
                        PCSAFT.SatV->dielc = PCSAFT.dielc_water(T);
                    } catch (const ValueError& ex) {
                        return 1e20;
                    }
                }

                if (PCSAFT.is_pure_or_pseudopure) {
                    PCSAFT.SatL->_rhomolar = PCSAFT.SatL->solver_rho_Tp(T, p, iphase_liquid);
                    vector<CoolPropDbl> fugcoef_l = PCSAFT.SatL->calc_fugacity_coefficients();

                    PCSAFT.SatV->_rhomolar = PCSAFT.SatV->solver_rho_Tp(T, p, iphase_gas);
                    vector<CoolPropDbl> fugcoef_v = PCSAFT.SatV->calc_fugacity_coefficients();
                    error += 100000 * pow(fugcoef_l[0] - fugcoef_v[0], 2.);
                }
                else {
                    if (PCSAFT.N > 1) {
                        bool reset_mole_fractions = false;
                        for (int i = 0; i < PCSAFT.N; i++) {
                            if (!ValidNumber(PCSAFT.SatL->mole_fractions[i]) || !ValidNumber(PCSAFT.SatV->mole_fractions[i])) {
                                reset_mole_fractions = true;
                            }
                        }
                        if (reset_mole_fractions) {
                            PCSAFT.SatL->mole_fractions = PCSAFT.mole_fractions;
                            PCSAFT.SatV->mole_fractions = PCSAFT.mole_fractions;
                        }
                    }

                    int itr = 0;
                    double dif = 10000.;
                    vector<CoolPropDbl> fugcoef_l(PCSAFT.N), fugcoef_v(PCSAFT.N);

                    double rhol, rhov, summ;
                    vector<CoolPropDbl> xv_old(PCSAFT.N);
                    double x_ions = 0.; // overall mole fraction of ions in the system
                    for (int i = 0; i < PCSAFT.N; i++) {
                        if (PCSAFT.components[i].getZ() != 0) {
                            x_ions += PCSAFT.mole_fractions[i];
                        }
                    }
                    while ((dif>1e-9) && (itr<100)) {
                        xv_old = PCSAFT.SatV->mole_fractions;
                        PCSAFT.SatL->_rhomolar = PCSAFT.SatL->solver_rho_Tp(T, p, iphase_liquid);
                        fugcoef_l = PCSAFT.SatL->calc_fugacity_coefficients();
                        PCSAFT.SatV->_rhomolar = PCSAFT.SatV->solver_rho_Tp(T, p, iphase_gas);
                        fugcoef_v = PCSAFT.SatV->calc_fugacity_coefficients();

                        if (PCSAFT._Q > 0.5) {
                            summ = 0.;
                            for (int i = 0; i < PCSAFT.N; i++) {
                                if (PCSAFT.components[i].getZ() == 0) {
                                    PCSAFT.SatL->mole_fractions[i] = fugcoef_v[i]*PCSAFT.SatV->mole_fractions[i]/fugcoef_l[i];
                                    summ += PCSAFT.SatL->mole_fractions[i];
                                }
                            }
                            for (int i = 0; i < PCSAFT.N; i++) {
                                if (PCSAFT.components[i].getZ() == 0) {
                                    PCSAFT.SatL->mole_fractions[i] = PCSAFT.SatL->mole_fractions[i]/summ*(((1-PCSAFT._Q) - x_ions)/(1-PCSAFT._Q)); // ensures that mole fractions add up to 1
                                    PCSAFT.SatV->mole_fractions[i] = (PCSAFT.mole_fractions[i] - (1-PCSAFT._Q)*PCSAFT.SatL->mole_fractions[i])/PCSAFT._Q; // if PCSAFT->_Q is close to zero then this equation behaves poorly, and that is why we use this if statement to switch the equation around
                                }
                                else {
                                    PCSAFT.SatL->mole_fractions[i] = PCSAFT.mole_fractions[i]/(1-PCSAFT._Q);
                                    PCSAFT.SatV->mole_fractions[i] = 0.;
                                }
                            }
                        }
                        else {
                            summ = 0.;
                            for (int i = 0; i < PCSAFT.N; i++) {
                                if (PCSAFT.components[i].getZ() == 0) {
                                    PCSAFT.SatV->mole_fractions[i] = fugcoef_l[i]*PCSAFT.SatL->mole_fractions[i]/fugcoef_v[i];
                                }
                                summ += PCSAFT.SatV->mole_fractions[i];
                            }
                            for (int i = 0; i < PCSAFT.N; i++) {
                                PCSAFT.SatV->mole_fractions[i] = PCSAFT.SatV->mole_fractions[i]/summ;
                                PCSAFT.SatL->mole_fractions[i] = (PCSAFT.mole_fractions[i] - (PCSAFT._Q)*PCSAFT.SatV->mole_fractions[i])/(1-PCSAFT._Q);
                            }
                        }

                        dif = 0;
                        for (int i = 0; i < PCSAFT.N; i++) {
                            dif += abs(PCSAFT.SatV->mole_fractions[i] - xv_old[i]);
                        }
                        itr += 1;
                    }

                    for (int i = 0; i < PCSAFT.N; i++) {
                        if (PCSAFT.components[i].getZ() == 0) {
                            error += pow(PCSAFT.SatL->mole_fractions[i]*fugcoef_l[i] - PCSAFT.SatV->mole_fractions[i]*fugcoef_v[i], 2.);
                        }
                        error += pow((PCSAFT.mole_fractions[i] - PCSAFT._Q*PCSAFT.SatV->mole_fractions[i] - (1-PCSAFT._Q)*PCSAFT.SatL->mole_fractions[i]), 2.);
                    }
                }

                if (!ValidNumber(error) || (PCSAFT.SatL->_rhomolar - PCSAFT.SatV->_rhomolar) < 1e-5) {
                    error = 1e20;
                }
            }
            return error;
        };
    };

    SolverTboilResid resid(*this, p);

    CoolPropDbl t_guess = _HUGE;
    double x_lo = _HUGE;
    double x_hi = _HUGE;

    double x_lbound = 1;
    double x_ubound = 1000;

    try {
        // scan through the range of temperatures to find a good initial guess
        int npts = 40;
        double err_min = 1e20;
        int ctr_increasing = 0; // keeps track of the number of steps where the error is increasing instead of decreasing
        for (int i = npts; i >= 0; i--) { // here we need to scan in the opposite direction (high T to low T) because a second, erroneous VLE occurs at lower temperatures. Reference: Privat R, Gani R, Jaubert JN. Are safe results obtained when the PC-SAFT equation of state is applied to ordinary pure chemicals?. Fluid Phase Equilibria. 2010 Aug 15;295(1):76-92.
            CoolPropDbl T_i = ((x_ubound - x_lbound) / (double)npts * i + x_lbound);
            double err = resid.call(T_i);

            if (err < err_min) {
                err_min = err;
                t_guess = T_i;
                x_lo = ((x_ubound - x_lbound) / (double)npts * (i - 1) + x_lbound);
                x_hi = ((x_ubound - x_lbound) / (double)npts * (i + 1) + x_lbound);
                ctr_increasing = 0;
            }
            else if (err_min < 1e20) {
                ctr_increasing += 1;
            }

            if (ctr_increasing > 2) { // this is necessary because PC-SAFT often gives a second, erroneous VLE at lower temperatures. Reference: Privat R, Gani R, Jaubert JN. Are safe results obtained when the PC-SAFT equation of state is applied to ordinary pure chemicals?. Fluid Phase Equilibria. 2010 Aug 15;295(1):76-92.
                break;
            }
        }

        if (t_guess == _HUGE) {
            throw SolutionError(format("A suitable initial guess for temperature could not be found for the PQ flash."));
        }
    }
    catch (const SolutionError& ex) {
        // scan through the range of temperatures to find a good initial guess
        int npts = 1000;
        double err_min = 1e20;
        int ctr_increasing = 0; // keeps track of the number of steps where the error is increasing instead of decreasing
        for (int i = npts; i >= 0; i--) { // here we need to scan in the opposite direction (high T to low T) because a second, erroneous VLE occurs at lower temperatures. Reference: Privat R, Gani R, Jaubert JN. Are safe results obtained when the PC-SAFT equation of state is applied to ordinary pure chemicals?. Fluid Phase Equilibria. 2010 Aug 15;295(1):76-92.
            CoolPropDbl T_i = ((x_ubound - x_lbound) / (double)npts * i + x_lbound);
            double err = resid.call(T_i);

            if (err < err_min) {
                err_min = err;
                t_guess = T_i;
                x_lo = ((x_ubound - x_lbound) / (double)npts * (i - 1) + x_lbound);
                x_hi = ((x_ubound - x_lbound) / (double)npts * (i + 1) + x_lbound);
                ctr_increasing = 0;
            }
            else if (err_min < 1e20) {
                ctr_increasing += 1;
            }

            if (ctr_increasing > 2) { // this is necessary because PC-SAFT often gives a second, erroneous VLE at lower temperatures. Reference: Privat R, Gani R, Jaubert JN. Are safe results obtained when the PC-SAFT equation of state is applied to ordinary pure chemicals?. Fluid Phase Equilibria. 2010 Aug 15;295(1):76-92.
                break;
            }
        }

        if (t_guess == _HUGE) {
            throw SolutionError(format("A suitable initial guess for temperature could not be found for the PQ flash."));
        }
    }

    CoolPropDbl T;
    try {
        T = BoundedSecant(resid, t_guess, x_lo, x_hi, 0.01*t_guess, 1e-8, 200);
    }
    catch (const SolutionError& ex) {
        T = BoundedSecant(resid, t_guess, x_lo, x_hi, 0.01*t_guess, 0.1, 200);
    }

    // Load the outputs
    PCSAFT._T = T;
    PCSAFT._rhomolar = 1/(PCSAFT._Q/PCSAFT.SatV->_rhomolar + (1 - PCSAFT._Q)/PCSAFT.SatL->_rhomolar);
    PCSAFT._phase = iphase_twophase;
}

CoolPropDbl PCSAFTBackend::solver_rho_Tp(CoolPropDbl T, CoolPropDbl p, phases phase){
    // Define the residual to be driven to zero
    class SolverRhoResid : public FuncWrapper1D
    {
    public:
        PCSAFTBackend &PCSAFT;
        CoolPropDbl T, p;

        SolverRhoResid(PCSAFTBackend &PCSAFT, CoolPropDbl T, CoolPropDbl p)
        : PCSAFT(PCSAFT), T(T), p(p){}
        CoolPropDbl call(CoolPropDbl rhomolar){
            CoolPropDbl peos = PCSAFT.update_DmolarT(rhomolar);
            double cost = (peos-p)/p;
            if (ValidNumber(cost)) {
                return cost;
            }
            else {
                return 1.0e20;
            }
        };
    };

    SolverRhoResid resid(*this, T, p);

    // split into grid and find bounds for each root
    vector<double> x_lo, x_hi;
    int num_pts = 25;
    double rho_guess = 1e-13;
    double rho_guess_prev = rho_guess;
    double err_prev = (update_DmolarT(reduced_to_molar(rho_guess, T)) - p) / p;
    for (int i = 0; i < num_pts; i++) {
        rho_guess = 0.7405 / (double)num_pts * i + 6e-3;
        double err = (update_DmolarT(reduced_to_molar(rho_guess, T)) - p) / p;
        if (err * err_prev < 0) {
            x_lo.push_back(rho_guess_prev);
            x_hi.push_back(rho_guess);
        }
        err_prev = err;
        rho_guess_prev = rho_guess;
    }

    // solve for appropriate root(s)
    double rho = _HUGE;
    double x_lo_molar, x_hi_molar;

    if (x_lo.size() == 1) {
        rho_guess = reduced_to_molar((x_lo[0] + x_hi[0]) / 2., T);
        x_lo_molar = reduced_to_molar(x_lo[0], T);
        x_hi_molar = reduced_to_molar(x_hi[0], T);
        rho = Brent(resid, x_lo_molar, x_hi_molar, DBL_EPSILON, 1e-8, 200);
    }
    else if (x_lo.size() <= 3 && !x_lo.empty()) {
        if ((phase == iphase_liquid) || (phase == iphase_supercritical_liquid)) {
            rho_guess = reduced_to_molar((x_lo.back() + x_hi.back()) / 2., T);
            x_lo_molar = reduced_to_molar(x_lo.back(), T);
            x_hi_molar = reduced_to_molar(x_hi.back(), T);
            rho = Brent(resid, x_lo_molar, x_hi_molar, DBL_EPSILON, 1e-8, 200);
        }
        else if ((phase == iphase_gas) || (phase == iphase_supercritical_gas) || (phase == iphase_supercritical)) {
            rho_guess = reduced_to_molar((x_lo[0] + x_hi[0]) / 40., T); // starting with a lower guess often provides better results
            x_lo_molar = reduced_to_molar(x_lo[0], T);
            x_hi_molar = reduced_to_molar(x_hi[0], T);
            rho = Brent(resid, x_lo_molar, x_hi_molar, DBL_EPSILON, 1e-8, 200);
        }
    }
    else if (x_lo.size() > 3) {
        // if multiple roots to check, then find the one with the minimum gibbs energy. Reference: Privat R, Gani R, Jaubert JN. Are safe results obtained when the PC-SAFT equation of state is applied to ordinary pure chemicals?. Fluid Phase Equilibria. 2010 Aug 15;295(1):76-92.
        double g_min = 1e60;
        for (int i = 0; i < x_lo.size(); i++) {
            rho_guess = reduced_to_molar((x_lo[i] + x_hi[i]) / 2., T);
            x_lo_molar = reduced_to_molar(x_lo[i], T);
            x_hi_molar = reduced_to_molar(x_hi[i], T);
            double rho_i = Brent(resid, x_lo_molar, x_hi_molar, DBL_EPSILON, 1e-8, 200);
            double rho_original = this->_rhomolar;
            this->_rhomolar = rho_i;
            double g_i = calc_gibbsmolar_residual();
            this->_rhomolar = rho_original;
            if (g_i < g_min) {
                g_min = g_i;
                rho = rho_i;
            }
        }
    }
    else {
        int num_pts = 25;
        double err_min = 1e40;
        double rho_min;
        for (int i = 0; i < num_pts; i++) {
            double rho_guess = 0.7405 / (double)num_pts * i + 1e-8;
            double err = (update_DmolarT(reduced_to_molar(rho_guess, T)) - p) / p;
            if (abs(err) < err_min) {
                err_min = abs(err);
                rho_min = reduced_to_molar(rho_guess, T);
            }
        }
        rho = rho_min;
    }

    return rho;
}

CoolPropDbl PCSAFTBackend::reduced_to_molar(CoolPropDbl nu, CoolPropDbl T) {
    vector<CoolPropDbl> d(N);
    CoolPropDbl summ = 0.;
    for (int i = 0; i < N; i++) {
        d[i] = components[i].getSigma()*(1-0.12*exp(-3*components[i].getU() / T));
        summ += mole_fractions[i]*components[i].getM()*pow(d[i],3.);
    }
    return 6/PI*nu/summ*1.0e30/N_AV;
}

CoolPropDbl PCSAFTBackend::calc_molar_mass(void){
    double summer = 0;
    for (unsigned int i = 0; i < N; ++i)
    {
        summer += mole_fractions[i] * components[i].molar_mass();
    }
    return summer;
}

vector<double> PCSAFTBackend::XA_find(vector<double> XA_guess, int ncA, vector<double> delta_ij, double den,
    vector<double> x) {
    /**Iterate over this function in order to solve for XA*/
    int n_sites = XA_guess.size()/ncA;
    double summ2;
    vector<CoolPropDbl> XA = XA_guess;

    for (int i = 0; i < ncA; i++) {
        for (int kout = 0; kout < n_sites; kout++) {
            summ2 = 0.;
            for (int j = 0; j < ncA; j++) {
                for (int kin = 0; kin < n_sites; kin++) {
                    if (kin != kout) {
                        summ2 += den*x[j]*XA_guess[j*n_sites+kin]*delta_ij[i*ncA+j];
                    }
                }
            }
            XA[i*n_sites+kout] = 1./(1.+summ2);
        }
    }

    return XA;
    }

vector<double> PCSAFTBackend::dXA_find(int ncA, int ncomp, vector<int> iA, vector<double> delta_ij,
    double den, vector<double> XA, vector<double> ddelta_dd, vector<double> x, int n_sites) {
    /**Solve for the derivative of XA with respect to density.*/
    Eigen::MatrixXd B(n_sites*ncA*ncomp, 1);
    Eigen::MatrixXd A = Eigen::MatrixXd::Zero(n_sites*ncA*ncomp, n_sites*ncA*ncomp);

    double sum1, sum2;
    int indx1, indx2;
    int indx4 = -1;
    int indx3 = -1;
    for (int i = 0; i < ncomp; i++) {
        indx1 = -1;
        if (find(iA.begin(), iA.end(), i) != iA.end()) {
            indx4 += 1;
        }
        for (int j = 0; j < ncA; j++) {
            for (int h = 0; h < n_sites; h++) {
                indx1 += 1;
                indx3 += 1;
                indx2 = -1;
                sum1 = 0;
                for (int k = 0; k < ncA; k++) {
                    for (int l = 0; l < n_sites; l++) {
                        indx2 += 1;
                        sum1 = sum1 + den*x[k]*(XA[indx2]*ddelta_dd[j*(ncA*ncomp)+k*(ncomp)+i]*((indx1+indx2)%2)); // (indx1+indx2)%2 ensures that A-A and B-B associations are set to zero
                        A(indx1+i*n_sites*ncA,indx2+i*n_sites*ncA) =
                        A(indx1+i*n_sites*ncA,indx2+i*n_sites*ncA) +
                        XA[indx1]*XA[indx1]*den*x[k]*delta_ij[j*ncA+k]*((indx1+indx2)%2);
                    }
                }

                sum2 = 0;
                if (find(iA.begin(), iA.end(), i) != iA.end()) {
                    for (int k = 0; k < n_sites; k++) {
                        sum2 = sum2 + XA[n_sites*(indx4)+k]*delta_ij[indx4*ncA+j]*((indx1+k)%2);
                    }
                }

                A(indx3,indx3) = A(indx3,indx3) + 1;
                B(indx3) = -1*XA[indx1]*XA[indx1]*(sum1 + sum2);
            }
        }
    }

    Eigen::MatrixXd solution = A.lu().solve(B); //Solves linear system of equations
    vector<double> dXA_dd(n_sites*ncA*ncomp);
    for (int i = 0; i < n_sites*ncA*ncomp; i++) {
        dXA_dd[i] = solution(i);
    }
    return dXA_dd;
}

vector<double> PCSAFTBackend::dXAdt_find(int ncA, vector<double> delta_ij, double den,
    vector<double> XA, vector<double> ddelta_dt, vector<double> x, int n_sites) {
    /**Solve for the derivative of XA with respect to temperature.*/
    Eigen::MatrixXd B = Eigen::MatrixXd::Zero(n_sites*ncA, 1);
    Eigen::MatrixXd A = Eigen::MatrixXd::Zero(n_sites*ncA, n_sites*ncA);

    double summ;
    int i_in, i_out = -1; // i_out is index of outer iteration loop (follows row of matrices)
    for (int i = 0; i < ncA; i++) {
        for (int ai = 0; ai < n_sites; ai++) {
            i_out += 1;
            i_in = -1; // index for summation loops
            summ = 0;
            for (int j = 0; j < ncA; j++) {
                for (int bj = 0; bj < n_sites; bj++) {
                    i_in += 1;
                    B(i_out) -= x[j]*XA[i_in]*ddelta_dt[i*ncA+j]*((i_in+i_out)%2); // (i_in+i_out)%2 ensures that A-A and B-B associations are set to zero
                    A(i_out,i_in) = x[j]*delta_ij[i*ncA+j]*((i_in+i_out)%2);
                    summ += x[j]*XA[i_in]*delta_ij[i*ncA+j]*((i_in+i_out)%2);
                }
            }
            A(i_out,i_out) = A(i_out,i_out) + pow(1+den*summ, 2.)/den;
        }
    }

    Eigen::MatrixXd solution = A.lu().solve(B); //Solves linear system of equations
    vector<double> dXA_dt(n_sites*ncA);
    for (int i = 0; i < n_sites*ncA; i++) {
        dXA_dt[i] = solution(i);
    }
    return dXA_dt;
}

double PCSAFTBackend::dielc_water(double t) {
    /**
    Return the dielectric constant of water at the given temperature.

    t : double
        Temperature (K)

    This equation was fit to values given in the reference. For temperatures
    from 263.15 to 368.15 K values at 1 bar were used. For temperatures from
    368.15 to 443.15 K values at 10 bar were used.

    Reference:
    D. G. Archer and P. Wang, “The Dielectric Constant of Water and Debye‐Hückel
    Limiting Law Slopes,” J. Phys. Chem. Ref. Data, vol. 19, no. 2, pp. 371–411,
    Mar. 1990.
    */
    double dielc;
    if (t < 263.15) {
        throw ValueError("The current function for the dielectric constant for water is only valid for temperatures above 263.15 K.");
    }
    else if (t <= 368.15) {
        dielc = 7.6555618295E-04*_T*_T - 8.1783881423E-01*_T + 2.5419616803E+02;
    }
    else if (t <= 443.15) {
        dielc = 0.0005003272124*_T*_T - 0.6285556029*_T + 220.4467027;
    }
    else {
        throw ValueError("The current function for the dielectric constant for water is only valid for temperatures less than 443.15 K.");
    }
    return dielc;
}

} /* namespace CoolProp */

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