1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
|
Subroutine get_lambda(Grad_K1CM, Vec_K1C, Vec_K1C_On_Evecs,
& Grad_K1C_On_Evecs, Hess_K1C, Evecs,
& AtmMass, Vec_K1C_Updated, Work,
& Coords_K1C, Stride, Delta, Nreals,
& Hess_Eval_Tol, Lamsearch_Tol,
& Binsearch_Tol)
Implicit Double Precision (A-H, O-Z)
C
Double Precision L_bound, Lamsearch_Tol, Mass, Lamda
C
Logical lamda_found, Bracket_end
C
Dimension Grad_K1CM(3*Nreals), Hess_K1C(3*Nreals, 3*Nreals),
& AtmMass(Nreals), Evecs(3*Nreals, 3*Nreals),
& Vec_K1C(3*Nreals), Vec_K1C_On_Evecs(3*Nreals),
& Grad_K1C_On_Evecs(3*Nreals), Work(27*Nreals*Nreals),
& Vec_K1C_Updated(3*Nreals), Coords_K1C(3*Nreals)
C
C Mass weigh the Hessian and obtain the eigenvectors.
C
#ifdef _DEBUG_LVL0
Write(6,*)
Write(6,*) "The Hessian"
Call output(Hess_K1C, 1, 3*Nreals, 1, 3*Nreals, 3*Nreals,
& 3*Nreals, 1)
#endif
Call Dcopy(9*Nreals*Nreals, Hess_K1C, 1,
& Work(18*nreals*nreals + 1), 1)
C
Do Ideg = 1, 3*Nreals
Do Jdeg = 1, 3*Nreals
Mass = DSQRT(AtmMass(1+(Ideg-1)/3)*AtmMass(1+(Jdeg-1)/3))
IF (Mass .lt. 1.0D-3) Then
Hess_K1C(Ideg, Jdeg) = 0.0D0
Else
Hess_K1C(Ideg, Jdeg) = Hess_K1C(Ideg, Jdeg)/Mass
Endif
Enddo
Enddo
C
#ifdef _DEBUG_LVL0
Write(6,*)
Write(6,*) "The mass weighted Hessian"
Call output(Hess_K1C, 1, 3*Nreals, 1, 3*Nreals, 3*Nreals,
& 3*Nreals, 1)
#endif
C
C Project the mass-weighted Hessian and diagonalize to define normal
C modes.
Call Projec_FC(Coords_K1C, Hess_K1C, AtmMass, Grad_K1CM, Work,
& WorK(9*Nreals*Nreals+1), 1.0D-8, Nreals, .True.,
& .True., .False.)
C
#ifdef _DEBUG_LVL0
NX = 3*Nreals
Write(6,"(a)") "The projected Hessian"
CALL OUTPUT(Hess_K1C, 1, NX, 1, Nx, Nx, Nx, 1)
#endif
Call Eig(Hess_K1C, Evecs, 3*Nreals, 3*Nreals, 0)
C
#ifdef _DEBUG_LVL0
Write(6,*)
NX = 3*Nreals
Write(6,"(a)") "The eigen vectors of the projected Hessian"
CALL OUTPUT(Evecs, 1, NX, 1, Nx, Nx, Nx, 1)
Write(6,"(a)") "The eigenvalues of the projected Hessian"
Write(6, "(4F17.13)") (Hess_K1C(I,I), I=1, NX)
#endif
C
#ifdef _DEBUG_LVL0
Write(6,*)
Write(6,*) "@-get_lambda Grad_K1CM (ZMAT order)"
Write(6, "(3F17.13)") (Grad_K1CM(i), i=1,3*Nreals)
Write(6,*)
Write(6,*) "@-get_lambda VEC_K1C (ZMAT order)"
Write(6, "(3F17.13)") (Vec_K1C(i), i=1,3*Nreals)
#endif
C
C Gradient along the eigenvectors of the Hessian
C
Do Imode = 1, 3*Nreals
Grad_K1C_on_Evecs(Imode) = Ddot(3*Nreals, Grad_K1CM, 1,
& Evecs(1, Imode), 1)
Vec_K1C_on_Evecs(Imode) = Ddot(3*Nreals, Vec_K1C, 1,
& Evecs(1, Imode), 1)
C
If (Dabs(Hess_K1C(Imode, Imode)) .LT. Hess_Eval_Tol) Then
Grad_K1C_on_Evecs(Imode) = 0.0D0
Vec_K1C_on_Evecs(Imode) = 0.0D0
Hess_K1C(Imode, Imode) = 0.0D0
Endif
C
Enddo
C
#ifdef _DEBUG_LVL0
Write(6,*)
Write(6,*) "The position and gradient vectors along e-vecs"
Write(6, "(F17.13)") (Grad_K1C_on_EvecS(I), I=1, 3*Nreals)
Write(6,*)
Write(6, "(F17.13)") (Vec_K1C_on_EvecS(I), I=1, 3*Nreals)
#endif
C
Imode = 3*Nreals
Eval_min = Hess_K1C(Imode, Imode)
Do While (Hess_K1C(Imode, Imode) .NE. 0.0D0 .OR. Imode .GT. 0)
Imode = Imode - 1
If (Hess_K1C(Imode, Imode) .NE. 0.0D0) Eval_min =
& Hess_K1C(Imode, Imode)
Enddo
C
#ifdef _DEBUG_LVL0
Write(6,*)
Write(6,*) "The lowest eiegnvalue of hess. and the tolerance"
Write(6, "(a,F17.13,a,F13.10)") "Eval_min =", Eval_min,
& " Hess_Eval_Tol =",Hess_Eval_Tol
#endif
C
C The solution to Eq. 26 im JPC, vol. 94, 5525, 1990 must be
C lower than the lowst eigenvalue of the Hessian. Another words
C Lambda < Eval_min.
C
halfs = 0.50D0*Stride
L_bound = Eval_min - (2.0D0)*Delta
U_bound = Eval_min - (1/2.0D0)*Delta
Halfs2 = halfs*halfs
niter = 1
Lbound_Tol = max((Eval_min)*Lamsearch_Tol,
& 10.0D0*Lamsearch_Tol)
Ubound_Tol = max(1.0D03, Abs(Eval_min))
Ubound_Tol = 1.0D03*Ubound_Tol
C
#ifdef _DEBUG_LVLM1
Write(6,*) "Lambda iterations"
Write(6,*) "Starting lower and upper limits"
Write(6,"(2F17.13)") L_bound, U_bound
#endif
C
Bracket_end = .FALSE.
Do while (.NOT. Bracket_end)
F_Lowr = 0.0D0
F_Uppr = 0.0D0
C
Do Imode = 1, 3*Nreals
Dnumer = Hess_K1C(Imode, Imode)*Vec_K1C_on_Evecs(Imode) -
& Grad_K1C_on_Evecs(Imode)
Denomi = Hess_K1C(Imode, Imode) - L_bound
F_Lowr = Dnumer**2/Denomi**2 + F_Lowr
Denomi = Hess_K1C(Imode, Imode) - U_bound
F_Uppr = Dnumer**2/Denomi**2 + F_Uppr
Enddo
#ifdef _DEBUG_LVLM1
Write(6,*) "The F_Lowr and F_Uppr"
Write(6,"(2F17.13)") F_Lowr, F_Uppr
#endif
C
F_Lowr = F_lowr - Halfs2
F_Uppr = F_Uppr - Halfs2
If (F_Lowr*F_Uppr .LT. 0.0D0) Then
Bracket_end = .TRUE.
Go to 10
Endif
C
niter = niter + 1
L_bound = Eval_min - (2.0D0)**niter*Delta
U_bound = Eval_min - (1.0D0/(2.0D0)**niter)*Delta
c
#ifdef _DEBUG_LVLM1
Write(6,*) "Lower and upper limits during its."
Write(6,"(2F17.13)") L_bound, U_bound
#endif
If (L_bound .LT. -Ubound_Tol .AND. ABS(Eval_min-U_bound) .LT.
& Lbound_Tol) Then
L_bound = -Ubound_Tol
U_bound = Eval_min - Lbound_Tol
Write(6, "(a,a)") "@-get_lambda Bracketing for lambda",
& "is failed"
Call Errex
Endif
Enddo
C
10 Continue
C
#ifdef _DEBUG_LVL0
Write(6,*)
Write(6,*) "The upper and lower bound for lambda"
Write(6, "(a,F17.13,a,F17.13)") " L_bound =", L_bound,
& " U_bound =",U_bound
#endif
C
C The binary search portion to extract the value in the range
C L_bound - U_bound.
C
Tmp_lamda = 0.0D0
Lambda_found = .False.
Niter = 0
Do while (.Not. lamda_found)
F_Lowr = 0.0D0
F_Uppr = 0.0D0
F_Midl = 0.0D0
Lamda = 0.50D0*(L_bound + U_bound)
#ifdef _DEBUG_LVLM1
Write(6,*) "The starting lambda"
Write(6, "(F17.13)") lamda
#endif
Do Imode = 1, 3*Nreals
Dnumer = Hess_k1C(Imode, Imode)*Vec_K1C_on_Evecs(Imode) -
& Grad_K1C_on_Evecs(Imode)
Denomi = Hess_K1C(Imode, Imode) - L_bound
F_Lowr = Dnumer**2/Denomi**2 + F_Lowr
Denomi = Hess_K1C(Imode, Imode) - U_bound
F_Uppr = Dnumer**2/Denomi**2 + F_Uppr
Denomi = Hess_K1C(Imode, Imode) - Lamda
F_Midl = Dnumer**2/Denomi**2 + F_Midl
Enddo
#ifdef _DEBUG_LVLM1
Write(6,*) "The FL, FU, FM during iterations"
Write(6,"(3F17.13)") F_Lowr, F_Uppr, F_Midl
#endif
F_Lowr = F_lowr - Halfs2
F_Uppr = F_Uppr - Halfs2
F_Midl = F_midl - Halfs2
C
If (Abs(Tmp_lamda -lamda) .LT. Binsearch_Tol)
& lamda_found = .True.
Niter = Niter + 1
If (Niter .gt. 1000) Then
Write(6, "(a,a)") "@-Get_lambda: The binary search",
& " for lambda failed!"
Call Errex
Endif
C
Tmp_lamda = Lamda
C
#ifdef _DEBUG_LVLM1
Write(6,*) "The FL, FU, FM post halfs2"
Write(6,"(3F17.13)") F_Lowr, F_Uppr, F_Midl
#endif
if (F_Lowr*F_Midl .LT. 0.0D0) U_bound = Lamda
If (F_Uppr*F_Midl .LT. 0.0D0) L_bound = Lamda
C
#ifdef _DEBUG_LVLM1
Write(6,*) "The L_bound and U_bound during iterations"
Write(6,"(2F17.13)") L_bound, U_bound
#endif
Enddo
If (Lamda .GT. Eval_min) Then
Write(6, "(a,a,2F10.5)")
& "@-Get_lambda: Warning, lambda is grater than the lowest",
& "eigenvalue", lamda, Eval_min
Call Errex
Endif
If (Abs(Lamda - Eval_min) .LT. Binsearch_Tol) Then
Write(6, "(a,a,F10.5)") "@-Get_lambda: Lamda is too close",
& "to the lowest eigenvalue", Lamda,
& Eval_min
Call Errex
Endif
#ifdef _DEBUG_LVL0
Write(6,*)
Write(6,*) "The value of lambda after binary search"
Write(6, "(a,F17.13)") " Lambda =", Lamda
#endif
C
C Make the Newton-Raphson Step
C
Call Dzero(Vec_K1C_Updated, 3*Nreals)
Do Imode = 1, 3*Nreals
If (Hess_K1C(Imode, Imode) .EQ. 0.0D0) Then
Do Jmode = 1, 3*Nreals
Vec_K1C_Updated(Jmode) = Vec_K1C_Updated(Jmode) +
& Hess_K1C(Imode, Imode)*
& Evecs(Jmode,Imode)
Enddo
Else
Denomi = (Lamda*Vec_K1C_on_Evecs(Imode) -
& Grad_K1C_on_Evecs(Imode))
Dnumer = Hess_K1C(Imode, Imode) - Lamda
Do Jmode = 1, 3*Nreals
Vec_K1C_Updated(Jmode) = Vec_K1C_Updated(Jmode) +
& (Denomi/Dnumer)*
& Evecs(Jmode,Imode)
Enddo
Endif
Enddo
C
#ifdef _DEBUG_LVL0
Write(6,*)
Write(6,*) "The M. W. updated vector (Vec_K1C_Updated)"
Write(6,"(3F17.13)") (Vec_K1C_Updated(i), i=1,3*Nreals)
#endif
C
C Eleminate mass weighing from the updated vector.
C
Ioff = 0
Do Iatom = 1, Nreals
Do Ixyz = 1, 3
Ioff = Ioff + 1
Vec_K1C_updated(Ioff) = Vec_K1C_updated(Ioff)/
& Dsqrt(AtmMass(Iatom))
Enddo
Enddo
C
Call Dcopy(9*Nreals*Nreals, Work(18*Nreals*Nreals+1), 1,
& Hess_K1C, 1)
C
#ifdef _DEBUG_LVLM1
Write(6,*)
Write(6,*) "The restored Hessian"
Call output(Hess_K1C, 1, 3*Nreals, 1, 3*Nreals, 3*Nreals,
& 3*Nreals, 1)
#endif
C
Return
End
|
