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
|
/**********************************************************************
XC_CA_LSDA.c:
XC_CA_LSDA is a subroutine to calculate an exchange-
correlation potential for a given density "den" by the local
spin density approximation, which is based on the original works
by Ceperly and Alder and parametrized by Perdew and Zunger.
Log of XC_CA_LSDA.c:
22/Nov/2001 Released by T.Ozaki
***********************************************************************/
#include <stdio.h>
#include <math.h>
#include "openmx_common.h"
void XC_CA_LSDA(double den0, double den1, double XC[2],int P_switch)
{
/****************************************************
P_switch:
0 \epsilon_XC (Exc, XC energy density)
1 \mu_XC (Vxc, XC potential)
2 \epsilon_XC - \mu_XC (Exc-Vxc)
****************************************************/
double dum,rs,coe,tden,min_den;
double Vxc,Ex,Ec,dEx,dEc,Exc,dExc;
double ExP,EcP,dExP,dEcP;
double ExF,EcF,dExF,dEcF;
double zeta,fzeta,dfzeta;
double tmp0,tmp1,tmp2;
double z0,z1,z02,z04,z12,z14;
/****************************************************
Non-relativisic formalism
****************************************************/
/****************************************************
total density (tden) = den0 + den1
zeta = (den0-den1)/tden
rs = pow(3.0/4.0/PI,1.0/3.0)*tden^{-1/3}
*****************************************************/
min_den = 1.0e-15;
tden = den0 + den1;
if (tden<min_den){
XC[0] = 0.0;
XC[1] = 0.0;
}
else{
zeta = (den0 - den1)/tden;
if (1.0<zeta) zeta = 1.0 - min_den;
if (zeta<-1.0) zeta = -1.0 + min_den;
coe = 0.6203504908994; /* pow(3.0/4.0/PI,1.0/3.0); */
if (tden<min_den)
rs = 62035.04908994; /* coe*pow(1.0e-15,-1.0/3.0); */
else
rs = coe*pow(tden,-0.3333333333333333333);
/*****************************************************
exchange energy density for the para magnetic state
ExP = -3/4*(3/pi)^(1/3)*tden^{1/3}
= -3/4*(9/4/pi/pi)^(1/3)/rs
= -0.458165293632163/rs
*****************************************************/
tmp0 = 0.458165293632163/rs;
ExP = -tmp0;
dExP = tmp0/rs;
/*****************************************************
exchange energy density for the ferro magnetic state
ExF = 2^(1/3)*ExP
= 1.25992104989487*ExP
*****************************************************/
ExF = 1.25992104989487*ExP;
dExF = 1.25992104989487*dExP;
/*****************************************************
correlation energy density for the para magnetic state
1<=rs
EcP = -0.1423/(1+1.0529*sqrt(rs) + 0.3334*rs)
0<=rs<=1
EcP = 0.0311*ln(rs)-0.048+0.0020*rs*ln(rs)-0.0116*rs
correlation energy density for the ferro magnetic state
1<=rs
EcF = -0.0843/(1+1.3981*sqrt(rs) + 0.2611*rs)
0<=rs<=1
EcF = 0.01555*ln(rs)-0.0269+0.0007*rs*ln(rs)-0.0048*rs
*****************************************************/
if (1.0<=rs){
tmp0 = sqrt(rs);
dum = (1.0 + 1.0529*tmp0 + 0.3334*rs);
tmp1 = 0.1423/dum;
EcP = -tmp1;
dEcP = tmp1/dum*(0.52645/tmp0 + 0.3334);
dum = (1.0 + 1.3981*tmp0 + 0.2611*rs);
tmp1 = 0.0843/dum;
EcF = -tmp1;
dEcF = tmp1/dum*(0.69905/tmp0 + 0.2611);
}
else{
tmp0 = log(rs);
EcP = -0.0480 + 0.0311*tmp0 + rs*(0.0020*tmp0 - 0.0116);
dEcP = 0.0311/rs + 0.0020*tmp0 - 0.0096;
EcF = -0.0269 + 0.01555*tmp0 + rs*(0.0007*tmp0 - 0.0048);
dEcF = 0.01555/rs + 0.0007*tmp0 - 0.0041;
}
if (P_switch==0){
/*****************************************************
z0 = (1 + zeta)^{4/3}
z1 = (1 - zeta)^{4/3}
fzeta = (z0 + z1 - 2)/(2*(2^{1/3}-1))
= 1.92366105093154*(z0 + z1 - 2)
*****************************************************/
z0 = pow(1.0+zeta,1.33333333333333333);
z1 = pow(1.0-zeta,1.33333333333333333);
fzeta = 1.92366105093154*(z0 + z1 - 2.0);
/*****************************************************
exchange-correration energy density
Ex = ExP + (ExF - ExP)*fzeta
Ec = EcP + (EcF - EcP)*fzeta
Exc = Ex + Ec
= ExP + EcP + (ExF + EcF - ExP - EcP)*fzeta
*****************************************************/
Exc = ExP + EcP + (ExF + EcF - ExP - EcP)*fzeta;
XC[0] = Exc;
XC[1] = Exc;
}
else if (P_switch==1){
/*****************************************************
z0 = (1 + zeta)^{1/3}
z1 = (1 - zeta)^{1/3}
fzeta = (z0^4 + z1^4 - 2)/(2*(2^{1/3}-1))
= 1.92366105093154*(z0^4 + z1^4 - 2)
dfzeta = 2.56488140124205*(z0 - z1)
*****************************************************/
z0 = pow(1.0+zeta,0.33333333333333333);
z1 = pow(1.0-zeta,0.33333333333333333);
z02 = z0*z0;
z04 = z02*z02;
z12 = z1*z1;
z14 = z12*z12;
fzeta = 1.92366105093154*(z04 + z14 - 2.0);
dfzeta = 2.56488140124205*(z0 - z1);
/*****************************************************
exchange-correration potential
Vxc+- = Exc + tden*dExc/drho +- dExc/dzeta*(1-+zeta)
Ex = ExP + (ExF - ExP)*fzeta
Ec = EcP + (EcF - EcP)*fzeta
Exc = Ex + Ec
= ExP + EcP + (ExF + EcF - ExP - EcP)*fzeta
dEx = dExP + (dExF - dExP)*fzeta
dEc = dEcP + (dEcF - dEcP)*fzeta
dExc = dEx + dEc
= dExP + dEcP + (dExF + dEcF - dExP - dEcP)*fzeta
tden*dExc/drho = -1/3*rs*dExc/drs
*****************************************************/
tmp0 = ExF + EcF - ExP - EcP;
Exc = ExP + EcP + tmp0*fzeta;
dExc = dExP + dEcP + (dExF + dEcF - dExP - dEcP)*fzeta;
Vxc = Exc - 0.33333333333333333333*rs*dExc;
XC[0] = Vxc + tmp0*( 1.0 - zeta)*dfzeta;
XC[1] = Vxc + tmp0*(-1.0 - zeta)*dfzeta;
}
else if (P_switch==2){
/*****************************************************
z0 = (1 + zeta)^{1/3}
z1 = (1 - zeta)^{1/3}
fzeta = (z0^4 + z1^4 - 2)/(2*(2^{1/3}-1))
= 1.92366105093154*(z0^4 + z1^4 - 2)
dfzeta = 2.56488140124205*(z0 - z1)
*****************************************************/
z0 = pow(1.0+zeta,0.33333333333333333);
z1 = pow(1.0-zeta,0.33333333333333333);
z02 = z0*z0;
z04 = z02*z02;
z12 = z1*z1;
z14 = z12*z12;
fzeta = 1.92366105093154*(z04 + z14 - 2.0);
dfzeta = 2.56488140124205*(z0 - z1);
/*****************************************************
Exc - Vxc
*****************************************************/
tmp0 = ExF + EcF - ExP - EcP;
Exc = ExP + EcP + tmp0*fzeta;
dExc = dExP + dEcP + (dExF + dEcF - dExP - dEcP)*fzeta;
Vxc = Exc - 0.33333333333333333333*rs*dExc;
tmp1 = 0.33333333333333333333*rs*dExc;
tmp2 = tmp0*dfzeta;
XC[0] = tmp1 - tmp2*( 1.0 - zeta);
XC[1] = tmp1 - tmp2*(-1.0 - zeta);
}
}
}
|
