HPhi++  3.1.0
Orthorhombic.cpp File Reference

Standard mode for the orthorhombic lattice. More...

#include "StdFace_vals.hpp"
#include "StdFace_ModelUtil.hpp"
#include <cstdlib>
#include <cstdio>
#include <cmath>
#include <complex>
#include <cstring>

Go to the source code of this file.

Functions

void StdFace_Orthorhombic (struct StdIntList *StdI)
 Setup a Hamiltonian for the Simple Orthorhombic lattice. More...
 

Detailed Description

Standard mode for the orthorhombic lattice.

Definition in file Orthorhombic.cpp.

Function Documentation

◆ StdFace_Orthorhombic()

void StdFace_Orthorhombic ( struct StdIntList *  StdI)

Setup a Hamiltonian for the Simple Orthorhombic lattice.

Author
Mitsuaki Kawamura (The University of Tokyo)

(1) Compute the shape of the super-cell and sites in the super-cell

(2) check & store parameters of Hamiltonian

(3) Set local spin flag (StdIntList::locspinflag) and the number of sites (StdIntList::nsite)

(4) Compute the upper limit of the number of Transfer & Interaction and malloc them.

(5) Set Transfer & Interaction

Parameters
[in,out]StdI

Definition at line 33 of file Orthorhombic.cpp.

References StdFace_Coulomb(), StdFace_FindSite(), StdFace_GeneralJ(), StdFace_Hopping(), StdFace_HubbardLocal(), StdFace_InitSite(), StdFace_InputCoulombV(), StdFace_InputHopp(), StdFace_InputSpin(), StdFace_InputSpinNN(), StdFace_MagField(), StdFace_MallocInteractions(), StdFace_NotUsed_c(), StdFace_NotUsed_d(), StdFace_NotUsed_i(), StdFace_NotUsed_J(), StdFace_PrintGeometry(), StdFace_PrintVal_c(), StdFace_PrintVal_d(), StdFace_PrintVal_i(), and StdFace_PrintXSF().

Referenced by StdFace_main().

36 {
37  int isite, jsite, ntransMax, nintrMax;
38  int iL, iW, iH, kCell;
39  FILE *fp;
40  std::complex<double> Cphase;
41  double dR[3];
42 
46  fp = fopen("lattice.xsf", "w");
47 
48  StdI->NsiteUC = 1;
49 
50  fprintf(stdout, " @ Lattice Size & Shape\n\n");
51 
52  StdFace_PrintVal_d("a", &StdI->a, 1.0);
53  StdFace_PrintVal_d("Wlength", &StdI->length[0], StdI->a);
54  StdFace_PrintVal_d("Llength", &StdI->length[1], StdI->a);
55  StdFace_PrintVal_d("Hlength", &StdI->length[2], StdI->a);
56  StdFace_PrintVal_d("Wx", &StdI->direct[0][0], StdI->length[0]);
57  StdFace_PrintVal_d("Wy", &StdI->direct[0][1], 0.0);
58  StdFace_PrintVal_d("Wz", &StdI->direct[0][2], 0.0);
59  StdFace_PrintVal_d("Lx", &StdI->direct[1][0], 0.0);
60  StdFace_PrintVal_d("Ly", &StdI->direct[1][1], StdI->length[1]);
61  StdFace_PrintVal_d("Lz", &StdI->direct[1][2], 0.0);
62  StdFace_PrintVal_d("Hx", &StdI->direct[2][0], 0.0);
63  StdFace_PrintVal_d("Hy", &StdI->direct[2][1], 0.0);
64  StdFace_PrintVal_d("Hz", &StdI->direct[2][2], StdI->length[1]);
65 
66  StdFace_PrintVal_d("phase0", &StdI->phase[0], 0.0);
67  StdFace_PrintVal_d("phase1", &StdI->phase[1], 0.0);
68  StdFace_PrintVal_d("phase2", &StdI->phase[2], 0.0);
69 
70  StdFace_InitSite(StdI, fp, 3);
71  StdI->tau[0][0] = 0.0; StdI->tau[0][1] = 0.0; ; StdI->tau[0][2] = 0.0;
75  fprintf(stdout, "\n @ Hamiltonian \n\n");
76  StdFace_NotUsed_d("K", StdI->K);
77  StdFace_PrintVal_d("h", &StdI->h, 0.0);
78  StdFace_PrintVal_d("Gamma", &StdI->Gamma, 0.0);
79 
80  if (strcmp(StdI->model, "spin") == 0 ) {
81  StdFace_PrintVal_i("2S", &StdI->S2, 1);
82  StdFace_PrintVal_d("D", &StdI->D[2][2], 0.0);
83  StdFace_InputSpinNN(StdI->J, StdI->JAll, StdI->J0, StdI->J0All, "J0");
84  StdFace_InputSpinNN(StdI->J, StdI->JAll, StdI->J1, StdI->J1All, "J1");
85  StdFace_InputSpinNN(StdI->J, StdI->JAll, StdI->J2, StdI->J2All, "J2");
86  StdFace_InputSpinNN(StdI->Jp, StdI->JpAll, StdI->J0p, StdI->J0pAll, "J0'");
87  StdFace_InputSpinNN(StdI->Jp, StdI->JpAll, StdI->J1p, StdI->J1pAll, "J1'");
88  StdFace_InputSpinNN(StdI->Jp, StdI->JpAll, StdI->J2p, StdI->J2pAll, "J2'");
89  StdFace_InputSpin(StdI->Jpp, StdI->JppAll, "J''");
90 
91  StdFace_NotUsed_d("mu", StdI->mu);
92  StdFace_NotUsed_d("U", StdI->U);
93  StdFace_NotUsed_c("t", StdI->t);
94  StdFace_NotUsed_c("t0", StdI->t0);
95  StdFace_NotUsed_c("t1", StdI->t1);
96  StdFace_NotUsed_c("t2", StdI->t2);
97  StdFace_NotUsed_c("t'", StdI->tp);
98  StdFace_NotUsed_c("t0'", StdI->t0p);
99  StdFace_NotUsed_c("t1'", StdI->t1p);
100  StdFace_NotUsed_c("t2'", StdI->t2p);
101  StdFace_NotUsed_c("t''", StdI->tpp);
102  StdFace_NotUsed_d("V", StdI->V);
103  StdFace_NotUsed_d("V0", StdI->V0);
104  StdFace_NotUsed_d("V1", StdI->V1);
105  StdFace_NotUsed_d("V'", StdI->Vp);
106  }/*if (strcmp(StdI->model, "spin") == 0 )*/
107  else {
108  StdFace_PrintVal_d("mu", &StdI->mu, 0.0);
109  StdFace_PrintVal_d("U", &StdI->U, 0.0);
110  StdFace_InputHopp(StdI->t, &StdI->t0, "t0");
111  StdFace_InputHopp(StdI->t, &StdI->t1, "t1");
112  StdFace_InputHopp(StdI->t, &StdI->t2, "t2");
113  StdFace_InputHopp(StdI->tp, &StdI->t0p, "t0'");
114  StdFace_InputHopp(StdI->tp, &StdI->t1p, "t1'");
115  StdFace_InputHopp(StdI->tp, &StdI->t2p, "t2'");
116  StdFace_PrintVal_c("tpp", &StdI->tpp, 0.0);
117  StdFace_InputCoulombV(StdI->V, &StdI->V0, "V0");
118  StdFace_InputCoulombV(StdI->V, &StdI->V1, "V1");
119  StdFace_InputCoulombV(StdI->V, &StdI->V2, "V2");
120  StdFace_InputCoulombV(StdI->Vp, &StdI->V0p, "V0'");
121  StdFace_InputCoulombV(StdI->Vp, &StdI->V1p, "V1'");
122  StdFace_InputCoulombV(StdI->Vp, &StdI->V2p, "V2'");
123  StdFace_PrintVal_d("Vpp", &StdI->Vpp, 0.0);
124 
125  StdFace_NotUsed_J("J0", StdI->J0All, StdI->J0);
126  StdFace_NotUsed_J("J1", StdI->J1All, StdI->J1);
127  StdFace_NotUsed_J("J2", StdI->J2All, StdI->J2);
128  StdFace_NotUsed_J("J0'", StdI->J0pAll, StdI->J0p);
129  StdFace_NotUsed_J("J1'", StdI->J1pAll, StdI->J1p);
130  StdFace_NotUsed_J("J2'", StdI->J2pAll, StdI->J2p);
131  StdFace_NotUsed_J("J0''", StdI->J0ppAll, StdI->J0pp);
132  StdFace_NotUsed_J("J1''", StdI->J1ppAll, StdI->J1pp);
133  StdFace_NotUsed_J("J2''", StdI->J2ppAll, StdI->J2pp);
134  StdFace_NotUsed_d("D", StdI->D[2][2]);
135 
136  if (strcmp(StdI->model, "hubbard") == 0 ) {
137  StdFace_NotUsed_i("2S", StdI->S2);
138  StdFace_NotUsed_J("J", StdI->JAll, StdI->J);
139  }/*if (strcmp(StdI->model, "hubbard") == 0 )*/
140  else {
141  StdFace_PrintVal_i("2S", &StdI->S2, 1);
142  StdFace_InputSpin(StdI->J, StdI->JAll, "J");
143  }/*if (model != "hubbard")*/
144 
145  }/*if (model != "spin")*/
146  fprintf(stdout, "\n @ Numerical conditions\n\n");
151  StdI->nsite = StdI->NsiteUC * StdI->NCell;
152  if (strcmp(StdI->model, "kondo") == 0 ) StdI->nsite *= 2;
153  StdI->locspinflag = (int *)malloc(sizeof(int) * StdI->nsite);
154 
155  if(strcmp(StdI->model, "spin") == 0 )
156  for (isite = 0; isite < StdI->nsite; isite++) StdI->locspinflag[isite] = StdI->S2;
157  else if(strcmp(StdI->model, "hubbard") == 0 )
158  for (isite = 0; isite < StdI->nsite; isite++) StdI->locspinflag[isite] = 0;
159  else
160  for (iL = 0; iL < StdI->nsite / 2; iL++) {
161  StdI->locspinflag[iL] = StdI->S2;
162  StdI->locspinflag[iL + StdI->nsite / 2] = 0;
163  }
167  if (strcmp(StdI->model, "spin") == 0 ) {
168  ntransMax = StdI->nsite * (StdI->S2 + 1/*h*/ + 2 * StdI->S2/*Gamma*/);
169  nintrMax = StdI->NCell * (StdI->NsiteUC/*D*/ + 3/*J*/ + 6/*J'*/ + 4/*J''*/)
170  * (3 * StdI->S2 + 1) * (3 * StdI->S2 + 1);
171  }
172  else {
173  ntransMax = StdI->NCell * 2/*spin*/ * (2 * StdI->NsiteUC/*mu+h+Gamma*/ + 6/*t*/ + 12/*t'*/ + 8/*t''*/);
174  nintrMax = StdI->NCell * (StdI->NsiteUC/*U*/ + 4 * (3/*V*/ + 6/*V'*/ + 4/*V''*/));
175 
176  if (strcmp(StdI->model, "kondo") == 0) {
177  ntransMax += StdI->nsite / 2 * (StdI->S2 + 1/*h*/ + 2 * StdI->S2/*Gamma*/);
178  nintrMax += StdI->nsite / 2 * (3 * StdI->S2 + 1) * (3 * StdI->S2 + 1);
179  }/*if (strcmp(StdI->model, "kondo") == 0)*/
180  }
181 
182  StdFace_MallocInteractions(StdI, ntransMax, nintrMax);
186  for (kCell = 0; kCell < StdI->NCell; kCell++){
187 
188  iW = StdI->Cell[kCell][0];
189  iL = StdI->Cell[kCell][1];
190  iH = StdI->Cell[kCell][2];
191  /*
192  (1) Local term
193  */
194  isite = kCell;
195  if (strcmp(StdI->model, "kondo") == 0 ) isite += StdI->NCell;
196 
197  if (strcmp(StdI->model, "spin") == 0 ) {
198  StdFace_MagField(StdI, StdI->S2, -StdI->h, -StdI->Gamma, isite);
199  StdFace_GeneralJ(StdI, StdI->D, StdI->S2, StdI->S2, isite, isite);
200  }/*if (strcmp(StdI->model, "spin") == 0 )*/
201  else {
202  StdFace_HubbardLocal(StdI, StdI->mu, -StdI->h, -StdI->Gamma, StdI->U, isite);
203  if (strcmp(StdI->model, "kondo") == 0 ) {
204  jsite = kCell;
205  StdFace_GeneralJ(StdI, StdI->J, 1, StdI->S2, isite, jsite);
206  StdFace_MagField(StdI, StdI->S2, -StdI->h, -StdI->Gamma, jsite);
207  }/*if (strcmp(StdI->model, "kondo") == 0 )*/
208  }
209  /*
210  (2) Nearest neighbor along W
211  */
212  StdFace_FindSite(StdI, iW, iL, iH, 1, 0, 0, 0, 0, &isite, &jsite, &Cphase, dR);
213 
214  if (strcmp(StdI->model, "spin") == 0 ) {
215  StdFace_GeneralJ(StdI, StdI->J0, StdI->S2, StdI->S2, isite, jsite);
216  }/*if (strcmp(StdI->model, "spin") == 0 )*/
217  else {
218  StdFace_Hopping(StdI, Cphase * StdI->t0, isite, jsite, dR);
219  StdFace_Coulomb(StdI, StdI->V0, isite, jsite);
220  }
221  /*
222  (3) Nearest neighbor along L
223  */
224  StdFace_FindSite(StdI, iW, iL, iH, 0, 1, 0, 0, 0, &isite, &jsite, &Cphase, dR);
225 
226  if (strcmp(StdI->model, "spin") == 0) {
227  StdFace_GeneralJ(StdI, StdI->J1, StdI->S2, StdI->S2, isite, jsite);
228  }
229  else {
230  StdFace_Hopping(StdI, Cphase * StdI->t1, isite, jsite, dR);
231  StdFace_Coulomb(StdI, StdI->V1, isite, jsite);
232  }
233  /*
234  (4) Nearest neighbor along H
235  */
236  StdFace_FindSite(StdI, iW, iL, iH, 0, 0, 1, 0, 0, &isite, &jsite, &Cphase, dR);
237 
238  if (strcmp(StdI->model, "spin") == 0) {
239  StdFace_GeneralJ(StdI, StdI->J2, StdI->S2, StdI->S2, isite, jsite);
240  }
241  else {
242  StdFace_Hopping(StdI, Cphase * StdI->t2, isite, jsite, dR);
243  StdFace_Coulomb(StdI, StdI->V2, isite, jsite);
244  }
245  /*
246  (5) Second nearest neighbor along +L+H
247  */
248  StdFace_FindSite(StdI, iW, iL, iH, 0, 1, 1, 0, 0, &isite, &jsite, &Cphase, dR);
249 
250  if (strcmp(StdI->model, "spin") == 0 ) {
251  StdFace_GeneralJ(StdI, StdI->J0p, StdI->S2, StdI->S2, isite, jsite);
252  }/*if (strcmp(StdI->model, "spin") == 0 )*/
253  else {
254  StdFace_Hopping(StdI, Cphase * StdI->t0p, isite, jsite, dR);
255  StdFace_Coulomb(StdI, StdI->V0p, isite, jsite);
256  }
257  /*
258  (6) Second nearest neighbor along +L-H
259  */
260  StdFace_FindSite(StdI, iW, iL, iH, 0, 1, -1, 0, 0, &isite, &jsite, &Cphase, dR);
261 
262  if (strcmp(StdI->model, "spin") == 0) {
263  StdFace_GeneralJ(StdI, StdI->J0p, StdI->S2, StdI->S2, isite, jsite);
264  }/*if (strcmp(StdI->model, "spin") == 0 )*/
265  else {
266  StdFace_Hopping(StdI, Cphase * StdI->t0p, isite, jsite, dR);
267  StdFace_Coulomb(StdI, StdI->V0p, isite, jsite);
268  }
269  /*
270  (7) Second nearest neighbor along +H+W
271  */
272  StdFace_FindSite(StdI, iW, iL, iH, 1, 0, 1, 0, 0, &isite, &jsite, &Cphase, dR);
273 
274  if (strcmp(StdI->model, "spin") == 0) {
275  StdFace_GeneralJ(StdI, StdI->J1p, StdI->S2, StdI->S2, isite, jsite);
276  }/*if (strcmp(StdI->model, "spin") == 0 )*/
277  else {
278  StdFace_Hopping(StdI, Cphase * StdI->t1p, isite, jsite, dR);
279  StdFace_Coulomb(StdI, StdI->V1p, isite, jsite);
280  }
281  /*
282  (8) Second nearest neighbor along +H-W
283  */
284  StdFace_FindSite(StdI, iW, iL, iH, -1, 0, 1, 0, 0, &isite, &jsite, &Cphase, dR);
285 
286  if (strcmp(StdI->model, "spin") == 0) {
287  StdFace_GeneralJ(StdI, StdI->J1p, StdI->S2, StdI->S2, isite, jsite);
288  }/*if (strcmp(StdI->model, "spin") == 0 )*/
289  else {
290  StdFace_Hopping(StdI, Cphase * StdI->t1p, isite, jsite, dR);
291  StdFace_Coulomb(StdI, StdI->V1p, isite, jsite);
292  }
293  /*
294  (9) Second nearest neighbor along +W+L
295  */
296  StdFace_FindSite(StdI, iW, iL, iH, 1, 1, 0, 0, 0, &isite, &jsite, &Cphase, dR);
297 
298  if (strcmp(StdI->model, "spin") == 0) {
299  StdFace_GeneralJ(StdI, StdI->J2p, StdI->S2, StdI->S2, isite, jsite);
300  }/*if (strcmp(StdI->model, "spin") == 0 )*/
301  else {
302  StdFace_Hopping(StdI, Cphase * StdI->t2p, isite, jsite, dR);
303  StdFace_Coulomb(StdI, StdI->V2p, isite, jsite);
304  }
305  /*
306  (10) Second nearest neighbor along +W-L
307  */
308  StdFace_FindSite(StdI, iW, iL, iH, 1, -1, 0, 0, 0, &isite, &jsite, &Cphase, dR);
309 
310  if (strcmp(StdI->model, "spin") == 0) {
311  StdFace_GeneralJ(StdI, StdI->J2p, StdI->S2, StdI->S2, isite, jsite);
312  }/*if (strcmp(StdI->model, "spin") == 0 )*/
313  else {
314  StdFace_Hopping(StdI, Cphase * StdI->t2p, isite, jsite, dR);
315  StdFace_Coulomb(StdI, StdI->V2p, isite, jsite);
316  }
317  /*
318  (11) Third nearest neighbor along +W+L+H
319  */
320  StdFace_FindSite(StdI, iW, iL, iH, 1, 1, 1, 0, 0, &isite, &jsite, &Cphase, dR);
321 
322  if (strcmp(StdI->model, "spin") == 0) {
323  StdFace_GeneralJ(StdI, StdI->Jpp, StdI->S2, StdI->S2, isite, jsite);
324  }/*if (strcmp(StdI->model, "spin") == 0 )*/
325  else {
326  StdFace_Hopping(StdI, Cphase * StdI->tpp, isite, jsite, dR);
327  StdFace_Coulomb(StdI, StdI->Vpp, isite, jsite);
328  }
329  /*
330  (12) Third nearest neighbor along -W+L+H
331  */
332  StdFace_FindSite(StdI, iW, iL, iH, -1, 1, 1, 0, 0, &isite, &jsite, &Cphase, dR);
333 
334  if (strcmp(StdI->model, "spin") == 0) {
335  StdFace_GeneralJ(StdI, StdI->Jpp, StdI->S2, StdI->S2, isite, jsite);
336  }/*if (strcmp(StdI->model, "spin") == 0 )*/
337  else {
338  StdFace_Hopping(StdI, Cphase * StdI->tpp, isite, jsite, dR);
339  StdFace_Coulomb(StdI, StdI->Vpp, isite, jsite);
340  }
341  /*
342  (13) Third nearest neighbor along +W-L+H
343  */
344  StdFace_FindSite(StdI, iW, iL, iH, 1, -1, 1, 0, 0, &isite, &jsite, &Cphase, dR);
345 
346  if (strcmp(StdI->model, "spin") == 0) {
347  StdFace_GeneralJ(StdI, StdI->Jpp, StdI->S2, StdI->S2, isite, jsite);
348  }/*if (strcmp(StdI->model, "spin") == 0 )*/
349  else {
350  StdFace_Hopping(StdI, Cphase * StdI->tpp, isite, jsite, dR);
351  StdFace_Coulomb(StdI, StdI->Vpp, isite, jsite);
352  }
353  /*
354  (14) Third nearest neighbor along +W+L-H
355  */
356  StdFace_FindSite(StdI, iW, iL, iH, 1, 1, -1, 0, 0, &isite, &jsite, &Cphase, dR);
357 
358  if (strcmp(StdI->model, "spin") == 0) {
359  StdFace_GeneralJ(StdI, StdI->Jpp, StdI->S2, StdI->S2, isite, jsite);
360  }/*if (strcmp(StdI->model, "spin") == 0 )*/
361  else {
362  StdFace_Hopping(StdI, Cphase * StdI->tpp, isite, jsite, dR);
363  StdFace_Coulomb(StdI, StdI->Vpp, isite, jsite);
364  }
365  }/*for (kCell = 0; kCell < StdI->NCell; kCell++)*/
366 
367  fclose(fp);
368  StdFace_PrintXSF(StdI);
369  StdFace_PrintGeometry(StdI);
370 }
void StdFace_InputHopp(std::complex< double > t, std::complex< double > *t0, const char *t0name)
Input hopping integral from the input file, if it is not specified, use the default value(0 or the is...
void StdFace_Coulomb(struct StdIntList *StdI, double V, int isite, int jsite)
Add onsite/offsite Coulomb term to the list StdIntList::Cinter and StdIntList::CinterIndx, and increase the number of them (StdIntList::NCinter).
void StdFace_GeneralJ(struct StdIntList *StdI, double J[3][3], int Si2, int Sj2, int isite, int jsite)
Treat J as a 3*3 matrix [(6S + 1)*(6S&#39; + 1) interactions].
void StdFace_InputSpin(double Jp[3][3], double JpAll, const char *Jpname)
Input spin-spin interaction other than nearest-neighbor.
void StdFace_PrintVal_d(const char *valname, double *val, double val0)
Print a valiable (real) read from the input file if it is not specified in the input file (=NaN)...
void StdFace_FindSite(struct StdIntList *StdI, int iW, int iL, int iH, int diW, int diL, int diH, int isiteUC, int jsiteUC, int *isite, int *jsite, std::complex< double > *Cphase, double *dR)
Find the index of transfer and interaction.
void StdFace_Hopping(struct StdIntList *StdI, std::complex< double > trans0, int isite, int jsite, double *dR)
Add Hopping for the both spin.
void StdFace_NotUsed_c(const char *valname, std::complex< double > val)
Stop HPhi if a variable (complex) not used is specified in the input file (!=NaN).
void StdFace_HubbardLocal(struct StdIntList *StdI, double mu0, double h0, double Gamma0, double U0, int isite)
Add intra-Coulomb, magnetic field, chemical potential for the itenerant electron. ...
void StdFace_NotUsed_J(const char *valname, double JAll, double J[3][3])
Stop HPhi if variables (real) not used is specified in the input file (!=NaN).
void StdFace_MagField(struct StdIntList *StdI, int S2, double h, double Gamma, int isite)
Add longitudinal and transvars magnetic field to the list.
void StdFace_PrintGeometry(struct StdIntList *StdI)
Print geometry of sites for the pos-process of correlation function.
void StdFace_PrintXSF(struct StdIntList *StdI)
Print lattice.xsf (XCrysDen format)
void StdFace_InputCoulombV(double V, double *V0, const char *V0name)
Input off-site Coulomb interaction from the input file, if it is not specified, use the default value...
void StdFace_MallocInteractions(struct StdIntList *StdI, int ntransMax, int nintrMax)
Malloc Arrays for interactions.
void StdFace_NotUsed_i(const char *valname, int val)
Stop HPhi if a variable (integer) not used is specified in the input file (!=2147483647, the upper limt of Int).
void StdFace_PrintVal_i(const char *valname, int *val, int val0)
Print a valiable (integer) read from the input file if it is not specified in the input file (=214748...
void StdFace_InitSite(struct StdIntList *StdI, FILE *fp, int dim)
Initialize the super-cell where simulation is performed.
void StdFace_PrintVal_c(const char *valname, std::complex< double > *val, std::complex< double > val0)
Print a valiable (complex) read from the input file if it is not specified in the input file (=NaN)...
void StdFace_InputSpinNN(double J[3][3], double JAll, double J0[3][3], double J0All, const char *J0name)
Input nearest-neighbor spin-spin interaction.
void StdFace_NotUsed_d(const char *valname, double val)
Stop HPhi if a variable (real) not used is specified in the input file (!=NaN).