void project()
{
ppp(xnxt,ynxt,h,wp);
rrr(xnxt,ynxt,h,wr);
ooo(xnxt,ynxt,h,wo);
jjj(xnxt,ynxt,h,wj);
eee(xnxt,ynxt,h,we);
ccc(xnxt,ynxt,h,wc);
ttt(xnxt,ynxt,h,wt);
}
/*============================================================================ */
void makematrix()
{
const double EZI0 = Sys.EZI0;
const double *ReEZI2 = Sys.ReEZI2;
const double *ImEZI2 = Sys.ImEZI2;
const double I = Sys.I;
const double NZI0 = Sys.NZI0;
const double HFI0 = Sys.HFI0;
const double *ReHFI2 = Sys.ReHFI2;
const double *ImHFI2 = Sys.ImHFI2;
const double Ib = Sys.Ib;
const double NZI0b = Sys.NZI0b;
const double HFI0b = Sys.HFI0b;
const double *ReHFI2b = Sys.ReHFI2b;
const double *ImHFI2b = Sys.ImHFI2b;
/* Diffusion parameters */
/*--------------------------------------------- */
const double DirTilt = Sys.DirTilt;
const double *d2psi = Sys.d2psi;
const double Rxx = Diff.Rxx;
const double Ryy = Diff.Ryy;
const double Rzz = Diff.Rzz;
const double ExchangeFreq = Diff.Exchange;
const double *xlk = Diff.xlk;
/* Lband = (lptmx>=2) ? lptmx*2 : 2; */
const int Lband = 8;
/* Kband = kptmx*2; */
const int Kband = 8;
/*========================================================== */
/* Loop variables */
int L1, jK1, K1, M1, pS1, qS1, pI1, qI1, pI1b, qI1b;
int L2, jK2, K2, M2, pS2, qS2, pI2, qI2, pI2b, qI2b;
int Ld, Ls, jKd, Kd, Ks, KK;
/* Min and max values for loop variables */
int K1max, M1max, qS1max, qI1max, qI1bmax;
int L2max, jK2min, K2min, K2max, M2min, M2max;
int pS2min, qS2min, qS2max, pI2min, qI2min, qI2max, pI2bmin, qI2bmin, qI2bmax;
int iRow = 0, iCol = 0, iElement = 0;
bool Potential = Diff.maxL>=0;
bool ExchangePresent = (ExchangeFreq!=0);
double IsoDiffKdiag, IsoDiffKm2, IsoDiffKp2, PotDiff;
bool diagRC, Ld2, diagLK, diagS, diagI;
double N_L, N_K, NormFactor;
double R_EZI2, R_HFI2, R_HFI2b, a1, a1b, g1, a2, a2b, g2;
int parityLK2, L, pId, qId, pIbd, qIbd;
double Term1, Term2, X;
int pd;
bool includeRank0;
double LiouvilleElement, GammaElement, t;
double d2jjj;
const bool RhombicDiff = (Rxx!=Ryy);
if (Display) {
mexPrintf(" Potential: %d Exchange: %d\n",Potential, ExchangePresent);
mexPrintf(" Lemax Lomax: %d %d %d\n",Lemax,Lomax);
mexPrintf(" jKmin Kmax Mmax: %d %d %d\n",jKmin,Kmax,Mmax);
mexPrintf(" pSmin; I Ib, pImax pIbmax: %d; %g %g, %d %d\n",pSmin,I,Ib,pImax,pIbmax);
mexPrintf(" NZI0a %0.3e HFI0a %0.3e\n",NZI0,HFI0);
mexPrintf(" NZI0b %0.3e HFI0b %0.3e\n",NZI0b,HFI0b);
mexPrintf(" ReHFI2a %0.3e %0.3e %0.3e %0.3e %0.3e\n",ReHFI2[0],ReHFI2[1],ReHFI2[2],ReHFI2[3],ReHFI2[4]);
mexPrintf(" ReHFI2b %0.3e %0.3e %0.3e %0.3e %0.3e\n",ReHFI2b[0],ReHFI2b[1],ReHFI2b[2],ReHFI2b[3],ReHFI2b[4]);
if (ImHFI2)
mexPrintf(" ReHFI2a %0.3e %0.3e %0.3e %0.3e %0.3e\n",ImHFI2[0],ImHFI2[1],ImHFI2[2],ImHFI2[3],ImHFI2[4]);
if (ImHFI2b)
mexPrintf(" ReHFI2b %0.3e %0.3e %0.3e %0.3e %0.3e\n",ImHFI2b[0],ImHFI2b[1],ImHFI2b[2],ImHFI2b[3],ImHFI2b[4]);
mexPrintf(" entering loops...\n");
}
/* All equation numbers refer to Meirovitch et al, J.Chem.Phys. 77 (1982) */
for (L1=0;L1<=Lemax;L1++) {
if (isodd(L1) && (L1>Lomax)) continue;
for (jK1=jKmin;jK1<=1;jK1+=2) {
K1max = mini(Kmax,L1);
for (K1=0;K1<=K1max;K1+=deltaK) {
if ((K1==0)&&(parity(L1)!=jK1)) continue;
/* Potential-independent part of diffusion operator */
/*-------------------------------------------------------- */
/* depends only on L and K and is diagonal in all except K */
IsoDiffKdiag = (Rxx+Ryy)/2*(L1*(L1+1))+K1*K1*(Rzz-(Rxx+Ryy)/2);
if (RhombicDiff) {
KK = K1-2;
IsoDiffKm2 = (Rxx-Ryy)/4*sqrt((L1-KK-1)*(L1-KK)*(L1+KK+1)*(L1+KK+2));
KK = K1+2;
IsoDiffKp2 = (Rxx-Ryy)/4*sqrt((L1+KK-1)*(L1+KK)*(L1-KK+1)*(L1-KK+2));
}
else {
IsoDiffKp2 = IsoDiffKm2 = 0;
}
/*-------------------------------------------------------- */
M1max = mini(Mmax,L1);
for (M1=-M1max;M1<=M1max;M1++) {
for (pS1=pSmin;pS1<=1;pS1++) {
qS1max = 1 - abs(pS1);
for (qS1=-qS1max;qS1<=qS1max;qS1+=2) {
for (pI1 = -pImax;pI1<=pImax;pI1++) {
qI1max = ((int)(2*I)) - abs(pI1);
for (qI1=-qI1max;qI1<=qI1max;qI1+=2) {
for (pI1b = -pIbmax;pI1b<=pIbmax;pI1b++) {
if ((MeirovitchSymm) && (DirTilt==0) && ((pI1+pI1b+pS1-M1)!=1)) continue; /* Eq. (A47), see Misra (A13) */
qI1bmax = ((int)(2*Ib)) - abs(pI1b);
for (qI1b=-qI1bmax;qI1b<=qI1bmax;qI1b+=2) {
iCol = iRow;
diagRC = true;
L2max = mini(Lemax,L1+Lband);
for (L2=L1;L2<=L2max;L2++) {
if (isodd(L2)&&(L2>Lomax)) continue;
Ld = L1 - L2; Ls = L1 + L2;
Ld2 = abs(Ld)<=2;
/* N_L normalisation factor, see after Eq. (A11) */
N_L = sqrt((double)((2.0*L1+1.0)*(2.0*L2+1.0)));
jK2min = (diagRC) ? jK1 : jKmin;
for (jK2=jK2min;jK2<=1;jK2+=2) {
jKd = jK1 - jK2;
K2max = mini(Kmax,L2);
K2min = (diagRC) ? K1 : 0;
for (K2=K2min;K2<=K2max;K2+=deltaK) {
if ((K2==0)&&(parity(L2)!=jK2)) continue;
diagLK = (L1==L2) && (K1==K2);
Kd = K1 - K2;
Ks = K1 + K2;
parityLK2 = parity(L2+K2);
/*---------------------------------------------------------------------- */
/* Pre-calculations for Liouville matrix elements, Eq. (A41) */
/*---------------------------------------------------------------------- */
/* R(mu=EZI,HFI;l=2), see Eq. (A42) and (A44) */
/*---------------------------------------------- */
R_EZI2 = 0; R_HFI2 = 0; R_HFI2b = 0;
if (Ld2) {
g1 = 0; a1 = 0; a1b = 0;
if (abs(Kd)<=2) {
const double coeff = jjj(L1,2,L2,K1,-Kd,-K2);
/*mexPrintf("[wigner3j(%d,%d,%d,%d,%d,%d) %g]\n",L1,2,L2,K1,-Kd,-K2,coeff); */
if (jK1==jK2) {
g1 = coeff*ReEZI2[Kd+2];
a1 = coeff*ReHFI2[Kd+2];
a1b = coeff*ReHFI2b[Kd+2];
}
else {
if (ImEZI2) g1 = coeff*ImEZI2[Kd+2]*jK1;
if (ImHFI2) a1 = coeff*ImHFI2[Kd+2]*jK1;
if (ImHFI2b) a1b = coeff*ImHFI2b[Kd+2]*jK1;
}
}
g2 = 0; a2 = 0; a2b = 0;
if (abs(Ks)<=2) {
const double coeff = jjj(L1,2,L2,K1,-Ks,K2);
if (jK1==jK2) {
g2 = coeff*ReEZI2[Ks+2];
a2 = coeff*ReHFI2[Ks+2];
a2b = coeff*ReHFI2b[Ks+2];
}
else {
if (ImEZI2) g2 = coeff*ImEZI2[Ks+2]*jK1;
if (ImHFI2) a2 = coeff*ImHFI2[Ks+2]*jK1;
if (ImHFI2b) a2b = coeff*ImHFI2b[Ks+2]*jK1;
}
}
R_EZI2 = g1 + jK2*parityLK2*g2;
R_HFI2 = a1 + jK2*parityLK2*a2;
R_HFI2b = a1b + jK2*parityLK2*a2b;
}
/* N_K(K_1,K_2) normalization factor, Eq. (A43) */
N_K = 1.0;
if (K1==0) N_K /= sqrt(2.0);
if (K2==0) N_K /= sqrt(2.0);
/* Normalisation prefactor in Eq.(A40) and Eq.(A41) */
NormFactor = N_L*N_K*parity(M1+K1);
/*---------------------------------------------------------------------- */
/*---------------------------------------------------------------------- */
/* Potential-dependent term of diffusion operator, Eq. (A40) */
/*---------------------------------------------------------------------- */
PotDiff = 0;
if (Potential) {
if ((abs(Ld)<=Lband)&&(parity(Ks)==1)&&
(jKd==0)&&(abs(Kd)<=Kband)&&(abs(Ks)<=Kband)) {
for (L=0; L<=Lband; L+=2) {
Term1 = 0;
if (Kd>=-L) {
X = xlk[(Kd+L)*(Diff.maxL+1) + L]; /* X^L_{K1-K2} */
if (X!=0)
Term1 = X * jjj(L1,L,L2,K1,-Kd,-K2);
}
Term2 = 0;
if (Ks<=L) {
X = xlk[(Ks+L)*(Diff.maxL+1) + L]; /* X^L_{K1+K2} */
if (X!=0)
Term2 = parityLK2*jK2* X * jjj(L1,L,L2,K1,-Ks,K2);
}
if (Term1 || Term2)
PotDiff += (Term1+Term2) * jjj(L1,L,L2,M1,0,-M1);
}
PotDiff *= NormFactor;
if (Display)
if (abs(PotDiff)>1e-10)
mexPrintf(" pot (%d,%d;%d,%d): %e\n",L1,L2,K1,K2,PotDiff);
}
}
/*---------------------------------------------------------------------- */
M2max = mini(Mmax,L2);
M2min = (diagRC) ? M1 : -M2max;
for (M2=M2min;M2<=M2max;M2++) {
int Md = M1 - M2;
bool diagLKM = diagLK && (jKd==0) && (Md==0);
/* Pre-compute 3j symbol in Eq. (A41) for l = 2 */
const double Liou3j = (Ld2) ? jjj(L1,2,L2,M1,-Md,-M2) : 0;
pS2min = (diagRC) ? pS1 : pSmin;
for (pS2=pS2min;pS2<=1;pS2++) {
int pSd = pS1 - pS2;
qS2max = 1 - abs(pS2);
qS2min = (diagRC) ? qS1 : -qS2max;
for (qS2=qS2min;qS2<=qS2max;qS2+=2) {
int qSd = qS1 - qS2;
diagS = (pS1==pS2) && (qS1==qS2);
pI2min = (diagRC) ? pI1 : -pImax;
for (pI2=pI2min;pI2<=pImax;pI2++) {
pId = pI1 - pI2;
qI2max = ((int)(2*I)) - abs(pI2);
qI2min = (diagRC) ? qI1 : -qI2max;
for (qI2=qI2min;qI2<=qI2max;qI2+=2) {
qId = qI1 - qI2;
pI2bmin = (diagRC) ? pI1b : -pIbmax;
for (pI2b=pI2bmin;pI2b<=pIbmax;pI2b++) {
if ((MeirovitchSymm) && (DirTilt==0) && ((pI2+pI2b+pS2-1)!=M2)) continue; /* Eq. (A47), see Misra (A13) */
pIbd = pI1b - pI2b;
qI2bmax = ((int)(2*Ib)) - abs(pI2b);
qI2bmin = (diagRC) ? qI1b : -qI2bmax;
for (qI2b=qI2bmin;qI2b<=qI2bmax;qI2b+=2) {
qIbd = qI1b - qI2b;
diagI = (pId==0) && (qId==0) && (pIbd==0) && (qIbd==0);
pd = pSd + pId + pIbd; /* not sure here; Misra (A11a) */
/*---------------------------------------------------------------------------- */
/* Matrix element of Liouville operator (Hamiltonian superoperator) */
/*---------------------------------------------------------------------------- */
/* For the rank-0 terms in (A41), the following simplifcations hold. */
/* - L1==L2, K1==K2 and M1==M2, otherwise the 3j are zero */
/* - The prefactor N_L*(-1)^(M1+K1) is canceled by the product of */
/* wigner3j(L,M;0,0;L,-M) within the sum and wigner3j(L,K;0,0;L,-K) from R. */
/* ( wigner3j(L,M;0,0;L,-M) = (-1)^(-M-L)/sqrt(2L+1) */
/* - The N_K factor is not needed because */
/* the l=0 ISTO components have not been transformed by */
/* the K-symmetrization. (following the formula, N_K is */
/* cancelled by R_0) */
LiouvilleElement = 0;
if (Ld2 && /* if |L1-L2| is not 2 or less, all 3j symbols in (A41) are zero */
(abs(Md)<=2) && /* otherwise first 3j symbol in (A41) is zero */
((DirTilt!=0) || (pd==Md)) && /* no director tilt -> d2 is diagonal, i.e. d2(DirTilt)(pd,Md) zero unless pd==Md */
(abs(pSd)<=1) && (abs(pId)<=1) && (abs(pIbd)<=1) && /* otherwise Clebsch-Gordans in (B7) and (B8) are zero */
(abs(pSd)==abs(qSd)) && (abs(pId)==abs(qId)) && (abs(pIbd)==abs(qIbd))) {
includeRank0 = diagLKM && /* l=0 in (A41) means L1=L2, M2-M1=0 and K2-K1=0 (3j symbols)*/
(pd==0); /* ? (A41): for DirTilt=0, d2 is diagonal, and M2-M1=0 means that pd=0 */
d2jjj = d2psi[(pd+2)+(Md+2)*5]*Liou3j; /* for l=2, see (A41) */
/* Electronic Zeeman interaction */
/*----------------------------------------- */
if (diagI) { /* diagonal in nuclear subspace, i.e. pId==0 and qId==0 for all nuclei */
/* Rank-2 term, Eq. (B7) and (B8) */
double C2, S_g;
if (pSd==0) { /* since pId==0, pSd and pSd+pId are equal */
C2 = +sqrt23; /* (112|000) */
S_g = pS1;
}
else {
C2 = +sqrt12; /* (112|-10-1), (112|101) */
S_g = -qSd/sqrt(2.0);
}
LiouvilleElement += NormFactor*d2jjj*R_EZI2*(C2*S_g);
/* Rank-0 term */
if (includeRank0) {
const double C0 = -sqrt13; /* (110|000) */
LiouvilleElement += EZI0*(C0*pS1);
}
}
/* Hyperfine interaction, nucleus 1 */
/*----------------------------------------- */
if ((I>0) && (pSd*pId==qSd*qId) && (pIbd==0) && (qIbd==0)) {
/* Compute Clebsch-Gordan coeffs and S_A from Eq. (B7) */
double C0, C2, S_A;
C0 = 0; C2 = 0; S_A = 0;
if (pId==0) {
if (pSd==0) {
S_A = (pS1*qI1+pI1*qS1)/2.0;
C0 = -sqrt13; /* (110|000) */
C2 = +sqrt23; /* (112|000) */
}
else {
S_A = -(pI1*pSd+qI1*qSd)/sqrt(8.0);
C0 = 0; /* no rank 0 term */
C2 = +sqrt12; /* (112|101), (112|-10-1) */
}
}
else {
int t = qI1*qId + pI1*pId;
double KI = sqrt(I*(I+1.0)-t*(t-2.0)/4.0);
if (pSd==0) {
S_A = -(pS1*pId+qS1*qId)*KI/sqrt(8.0);
C0 = 0; /* no rank 0 term */
C2 = +sqrt12; /* (112|011), (112|0-1-1) */
}
else {
S_A = pSd*qId*KI/2.0;
C0 = +sqrt13; /* (110|1-10), (110|-110) */
if (pSd+pId==0)
C2 = +sqrt(1.0/6.0); /* (112|1-10), (112|-110) */
else
C2 = +1.0; /* (112|112), (112|-1-1-2) */
}
}
/* Rank-2 term, Eq. (A41) */
LiouvilleElement += NormFactor*d2jjj*R_HFI2 * (C2*S_A); /* Eq. (A41) and (B7)*/
/* Rank-0 term */
if (includeRank0)
LiouvilleElement += HFI0*(C0*S_A);
}
/* Hyperfine interaction, nucleus b */
/*----------------------------------------- */
if ((Ib>0) && (pSd*pIbd==qSd*qIbd) && (pId==0) && (qId==0)) {
/* Compute Clebsch-Gordan coeffs and S_A from Eq. (B7) */
double C0, C2, S_A;
C0 = 0; C2 = 0; S_A = 0;
if (pIbd==0) {
if (pSd==0) {
S_A = (pS1*qI1b+pI1b*qS1)/2.0;
C0 = -sqrt13; /* (110|000) */
C2 = +sqrt23; /* (112|000) */
}
else {
S_A = -(pI1b*pSd+qI1b*qSd)/sqrt(8.0);
C0 = 0; /* no rank 0 term */
C2 = +sqrt12; /* (112|101), (112|-10-1) */
}
}
else {
int t = qI1b*qIbd + pI1b*pIbd;
double KIb = sqrt(Ib*(Ib+1.0)-t*(t-2.0)/4.0);
if (pSd==0) {
S_A = -(pS1*pIbd+qS1*qIbd)*KIb/sqrt(8.0);
C0 = 0; /* no rank 0 term */
C2 = +sqrt12; /* (112|011), (112|0-1-1) */
}
else {
S_A = pSd*qIbd*KIb/2.0;
C0 = +sqrt13; /* (110|1-10), (110|-110) */
if (pSd+pIbd==0)
C2 = +sqrt(1.0/6.0); /* (112|1-10), (112|-110) */
else
C2 = +1.0; /* (112|112), (112|-1-1-2) */
}
}
/* Rank-2 term, Eq. (A41) */
LiouvilleElement += NormFactor*d2jjj*R_HFI2b * (C2*S_A); /* Eq. (A41) and (B7)*/
/* Rank-0 term */
if (includeRank0)
LiouvilleElement += HFI0b*(C0*S_A);
}
/* Nuclear Zeeman interaction */
/*----------------------------------------- */
/* has only a rank-0 component */
if (diagS && diagI && includeRank0) {
const double C0 = -sqrt13; /* (110|000) */
LiouvilleElement += NZI0*C0*pI1;
LiouvilleElement += NZI0b*C0*pI1b;
}
}
/*------------------------------------------- */
/*------------------------------------------------------ */
/* Matrix element of diffusion superoperator */
/*------------------------------------------------------ */
GammaElement = 0;
if (diagS && diagI) { /* all potential terms are diagonal in the spin space */
/* Potential-independent terms, Eq. (A15) */
if ((Ld==0) && (Md==0) && (jKd==0)) {
if (Kd==0) GammaElement += IsoDiffKdiag;
else if (Kd==+2) GammaElement += IsoDiffKm2/N_K;
else if (Kd==-2) GammaElement += IsoDiffKp2/N_K;
}
/* Potential-dependent terms, Eq. (A40) */
if (Potential) {
if ((Md==0) && (jKd==0))
GammaElement += PotDiff;
}
}
/* Exchange term */
if (ExchangePresent) {
if ((pSd==0) && (pId==0) && (pIbd==0) && diagLKM) {
t = 0;
if ((qId==0) && (qIbd==0) && (qSd==0)) t += 1.0;
if ((qId==0) && (qIbd==0) && (pS1==0)) t -= 0.5;
if ((pI1==0) && (pI1b==0) && (qSd==0)) t -= 1.0/(2.0*I+1)/(2.0*Ib+1);
GammaElement += t*ExchangeFreq;
}
}
/*------------------------------------------- */
/*------------------------------------------- */
/* Store element values and indices */
/*------------------------------------------- */
if ((GammaElement!=0) || (LiouvilleElement!=0)) {
MatrixRe[iElement] = GammaElement;
MatrixIm[iElement] = -LiouvilleElement;
ridx[iElement] = iRow;
cidx[iElement] = iCol;
iElement++;
if (!diagRC) {
MatrixRe[iElement] = GammaElement;
MatrixIm[iElement] = -LiouvilleElement;
ridx[iElement] = iCol;
cidx[iElement] = iRow;
iElement++;
}
if (iElement>=maxElements)
mexErrMsgTxt("Number of non-zero elements too large. Increase Opt.Allocation(1).");
}
/*------------------------------------------- */
iCol++;
diagRC = false;
if (iCol>=maxRows)
mexErrMsgTxt("Dimension too large. Increase Opt.Allocation(2).");
} /* qI2b */
} /* pI2b */
} /* qI2 */
} /* pI2 */
} /* qS2 */
} /* pS2 */
} /* M2 */
} /* K2 */
} /* jK2 */
} /* L2 */ /* all column index loops */
iRow++;
if (iRow>=maxRows)
mexErrMsgTxt("Dimension too large. Increase Opt.Allocation(2).");
} /* qI1b */
} /* pI1b */
} /* qI1 */
} /* pI1 */
} /* qS1 */
} /* pS1 */
} /* M1 */
} /* K1 */
} /* jK1 */
} /* L1; all row index loops */
nRows = iRow;
nElements = iElement;
if (Display)
mexPrintf(" %d elements, %d rows;\n",nElements,nRows);
}
