Theor Chem Acc (2015) 134:148
1 3
percent). It is quite clear from Fig. 3 that the errors for the
molecules containing second-row elements are considerably higher than for most of the molecules composed of H,
C, N, O and F atoms. Notable exceptions are the N 2 and
N 2 O molecules.
The performance of the method is signifi cantly better for orbitals obtained by nonlocal exchange: RHF and
RSHLDA. In the latter case the long-range exchange is
nonlocal, and only the short-range exchange is described
by a short-range functional. The best performance has been
achieved by the RSHLDA[M] method. In contrast to the
pure DFA calculations, the matrix and operator algebra
methods differ for the RHF and RSH methods signifi cantly:
the mean absolute error and the standard deviation of the
Fig. 3 Percentage errors in calculated molecular C 6 coeffi cients
obtained with different methods using the matrix algebra approach
Table 1 Molecular C 6 coeffi cients from the dipolar oscillator orbital method, using LDA, PBE, RHF and sr-LDA/lr-RHF determinants in augcc-pVTZ basis set and Boys localized orbitals
The matrix elements are calculated with a matrix algebra [M] and operator algebra [O] approach, respectively. All results are in a.u.
Values highlighted as bold indicate the best agreement with experiment
Molecules
Ref.
LDA[M]
LDA[O]
PBE[M]
PBE[O]
RHF[M]
RHF[O]
RSHLDA[M]
RSHLDA[O]
H 2
12.1
18.4
18.4
17.2
17.2
9.6
16.2
10.6
16.9
HF
19.0
20.7
21.5
20.7
21.6
12.4
17.2
14.9
19.3
H 2 O
45.3
55.6
56.5
55.1
56.1
31.9
45.4
36.4
48.9
N 2
73.3
118.3
118.8
117.6
118.0
75.2
109.0
81.1
110.9
CO
81.4
120.4
120.9
119.1
119.7
70.9
104.1
79.1
109.6
NH 3
89.0
118.8
119.9
116.7
117.8
66.3
98.6
72.6
102.0
CH 4
129.7
192.4
193.0
185.9
186.5
105.3
163.5
114.5
168.2
HCl
130.4
208.9
211.8
205.1
208.5
121.5
186.3
126.7
184.8
CO 2
158.7
234.9
237.0
233.2
235.4
130.9
184.0
153.3
202.8
H 2 CO
165.2
205.7
207.2
202.8
204.3
115.2
168.3
129.4
178.4
N 2 O
184.9
317.1
319.3
315.3
317.5
178.9
252.3
206.0
274.7
C 2 H 2
204.1
343.5
345.2
340.4
342.2
206.4
316.1
209.7
306.5
HBr
216.6
303.3
325.9
302.5
325.4
188.3
293.5
188.2
282.7
H 2 S
216.8
392.8
397.7
382.8
388.2
213.0
339.2
220.5
335.5
CH 3 OH
222.0
303.8
305.3
297.0
298.5
166.8
246.2
187.2
260.2
SO 2
294.0
542.6
554.9
539.8
552.5
284.9
416.7
329.5
456.2
C 2 H 4
300.2
466.1
467.8
456.7
458.4
266.3
406.4
279.4
405.8
CH 3 NH 2
303.8
440.6
442.3
429.8
431.4
240.2
360.3
263.2
373.3
SiH 4
343.9
639.6
655.3
598.4
613.9
280.0
484.6
310.9
513.0
C 2 H 6
381.9
579.2
580.9
560.9
562.5
313.5
480.9
341.2
495.5
Cl 2
389.2
727.4
735.4
714.3
723.7
401.3
607.5
424.3
612.2
CH 3 CHO
401.7
627.5
630.4
613.3
616.0
333.2
493.6
372.6
521.7
COS
402.2
845.7
855.8
843.9
853.7
456.3
689.2
495.7
713.5
CH 3 OCH 3
534.1
781.4
784.3
758.4
761.1
415.2
619.4
464.8
654.3
C 3 H 6
662.1
1045.9
1049.4
1018.6
1021.8
571.2
868.2
609.8
881.2
CS 2
871.1
2099.4
2119.5
2073.4
2094.5
1085.6
1658.0
1144.0
1686.2
CCl 4
2024.1
3831.3
3861.0
3750.4
3784.0
2004.8
3007.0
2135.5
3051.9
MAD%E
59.84
61.69
56.71
58.62
15.22
33.75
11.84
37.60
STD%E
28.08
28.25
27.76
27.98
9.88
21.15
7.24
20.86
CSSD%E
67.14
68.92
64.11
65.97
18.39
40.37
14.07
43.62
107
Reprinted from the journal
1 3
percent). It is quite clear from Fig. 3 that the errors for the
molecules containing second-row elements are considerably higher than for most of the molecules composed of H,
C, N, O and F atoms. Notable exceptions are the N 2 and
N 2 O molecules.
The performance of the method is signifi cantly better for orbitals obtained by nonlocal exchange: RHF and
RSHLDA. In the latter case the long-range exchange is
nonlocal, and only the short-range exchange is described
by a short-range functional. The best performance has been
achieved by the RSHLDA[M] method. In contrast to the
pure DFA calculations, the matrix and operator algebra
methods differ for the RHF and RSH methods signifi cantly:
the mean absolute error and the standard deviation of the
Fig. 3 Percentage errors in calculated molecular C 6 coeffi cients
obtained with different methods using the matrix algebra approach
Table 1 Molecular C 6 coeffi cients from the dipolar oscillator orbital method, using LDA, PBE, RHF and sr-LDA/lr-RHF determinants in augcc-pVTZ basis set and Boys localized orbitals
The matrix elements are calculated with a matrix algebra [M] and operator algebra [O] approach, respectively. All results are in a.u.
Values highlighted as bold indicate the best agreement with experiment
Molecules
Ref.
LDA[M]
LDA[O]
PBE[M]
PBE[O]
RHF[M]
RHF[O]
RSHLDA[M]
RSHLDA[O]
H 2
12.1
18.4
18.4
17.2
17.2
9.6
16.2
10.6
16.9
HF
19.0
20.7
21.5
20.7
21.6
12.4
17.2
14.9
19.3
H 2 O
45.3
55.6
56.5
55.1
56.1
31.9
45.4
36.4
48.9
N 2
73.3
118.3
118.8
117.6
118.0
75.2
109.0
81.1
110.9
CO
81.4
120.4
120.9
119.1
119.7
70.9
104.1
79.1
109.6
NH 3
89.0
118.8
119.9
116.7
117.8
66.3
98.6
72.6
102.0
CH 4
129.7
192.4
193.0
185.9
186.5
105.3
163.5
114.5
168.2
HCl
130.4
208.9
211.8
205.1
208.5
121.5
186.3
126.7
184.8
CO 2
158.7
234.9
237.0
233.2
235.4
130.9
184.0
153.3
202.8
H 2 CO
165.2
205.7
207.2
202.8
204.3
115.2
168.3
129.4
178.4
N 2 O
184.9
317.1
319.3
315.3
317.5
178.9
252.3
206.0
274.7
C 2 H 2
204.1
343.5
345.2
340.4
342.2
206.4
316.1
209.7
306.5
HBr
216.6
303.3
325.9
302.5
325.4
188.3
293.5
188.2
282.7
H 2 S
216.8
392.8
397.7
382.8
388.2
213.0
339.2
220.5
335.5
CH 3 OH
222.0
303.8
305.3
297.0
298.5
166.8
246.2
187.2
260.2
SO 2
294.0
542.6
554.9
539.8
552.5
284.9
416.7
329.5
456.2
C 2 H 4
300.2
466.1
467.8
456.7
458.4
266.3
406.4
279.4
405.8
CH 3 NH 2
303.8
440.6
442.3
429.8
431.4
240.2
360.3
263.2
373.3
SiH 4
343.9
639.6
655.3
598.4
613.9
280.0
484.6
310.9
513.0
C 2 H 6
381.9
579.2
580.9
560.9
562.5
313.5
480.9
341.2
495.5
Cl 2
389.2
727.4
735.4
714.3
723.7
401.3
607.5
424.3
612.2
CH 3 CHO
401.7
627.5
630.4
613.3
616.0
333.2
493.6
372.6
521.7
COS
402.2
845.7
855.8
843.9
853.7
456.3
689.2
495.7
713.5
CH 3 OCH 3
534.1
781.4
784.3
758.4
761.1
415.2
619.4
464.8
654.3
C 3 H 6
662.1
1045.9
1049.4
1018.6
1021.8
571.2
868.2
609.8
881.2
CS 2
871.1
2099.4
2119.5
2073.4
2094.5
1085.6
1658.0
1144.0
1686.2
CCl 4
2024.1
3831.3
3861.0
3750.4
3784.0
2004.8
3007.0
2135.5
3051.9
MAD%E
59.84
61.69
56.71
58.62
15.22
33.75
11.84
37.60
STD%E
28.08
28.25
27.76
27.98
9.88
21.15
7.24
20.86
CSSD%E
67.14
68.92
64.11
65.97
18.39
40.37
14.07
43.62
107
Reprinted from the journal