Theor Chem Acc (2015) 134:148
1 3
In the present work, the exchange contribution, which
is present only in the case of Hartree–Fock or hybrid density functional fockians and which would give rise to nonconventional two-electron integrals, is not treated explicitly. Possible approximate solutions for this problem will
be discussed in forthcoming works. Although we do not
use the elements of the off-diagonal ( i = j ) blocks of the
POO fock operator, for the sake of completeness we give
its expression:
To derive this, instead of the double commutator of
Eq. 73 , one needs to consider the product of the commutator with a coordinate operator
where ˆ
T is the kinetic energy operator, and [ˆ r β , ˆ
T ] = ˆ
∇ β .
References
1. Perdew JP, Schmidt K (2001) In: van Doren V, van Alsenoy K,
Geerlings P (eds) AIP conference proceedings (AIP), pp 1–20
2. Eichkorn K, Treutler O, Ohm H, Haser M, Ahlrichs R (1995)
Chem Phys Lett 240:283
3. Whitten JL (1973) J Chem Phys 58:4496
4. Roeggen I, Johansen T (2008) J Chem Phys 128:194107
5. Beebe NHF, Linderberg J (1977) Int J Quantum Chem 12:683
6. Harl J, Kresse G (2008) Phys Rev B 77:045136
7. Nguyen H-V, de Gironcoli S (2009) Phys Rev B 79:205114
8. Wilson HF, Lu D, Gygi F, Galli G (2009) Phys Rev B 79:245106
9. Wilson HF, Gygi F, Galli G (2008) Phys Rev B 78:113303
10. Rocca D (2014) J Chem Phys 140:18A501
11. Surján PR (2005) Chem Phys Lett 406:318
12. Johnson ER, Becke AD (2005) J Chem Phys 123:024101
13. Johnson ER, Mackie ID, DiLabio GA (2009) J Phys Org Chem
22:1127
14. Grimme S, Antony J, Ehrlich S, Krieg H (2010) J Chem Phys
132:154104
15. Grimme S (2011) Wires Comput Mol Sci 1:211
16. Tkatchenko A, Scheffl er M (2009) Phys Rev Lett 102:073005
(81)
i α | ˆ
f |i β = 1
2 δ αβ − 1
2
occ
m
im|
ˆ
r α − ˆ
r
α
w
r, r
ˆ
r β − ˆ
r
β
|mi
+ 1
2
occ
m
i|ˆ r α ˆ
r β |mf mi + f im m|ˆ r α ˆ
r β |i
−
occ
mn
i|ˆ r α |mf mn n|ˆ r β |i.
(82)
i α | ˆ
f |j β = −−i |ˆ r α ˆ
∇ β |j +
occ
m
im|ˆ r α w
r, r
ˆ
r β − ˆ
r
β
|mj
+
occ
m
i |ˆ r α ˆ
r β |m f mi
−
occ
mn
i |ˆ r α |m f mn n|ˆ r β |i.
(83)
ˆ
r α
ˆ
r β , ˆ
f
= ˆ
r α
ˆ
r β , ˆ
T
− ˆ
r α
ˆ
r β , ˆ
K
,
17. DiStasio RA Jr, Gobre VV, Tkatchenko A (2014) J Phys Condens Matter 26:213202
18. Reilly AM, Tkatchenko A (2015) Chem Sci 6:3289
19. Surján PR (1989) In: Maksic ZB (ed) Theoretical models of
chemical bonding, vol 2. Springer, Heidelberg, pp 205–256
20. Høyvik I-M, Jansík B, Jørgensen P (2012) J Chem Phys
137:224114
21. Pulay P (1983) Chem Phys Lett 100:151
22. Pulay P, Saebø S (1986) Theor Chim Acta 69:357
23. Boughton JW, Pulay P (1993) J Comput Chem 14:736
24. Foster JM, Boys SF (1960a) Rev Mod Phys 32:300
25. Foster JM, Boys SF (1960b) Rev Mod Phys 32:303
26. Boys SF (1966) In: Löwdin PO (ed) Quantum theory of atoms,
molecules, and the solid state, a tribute to John C. Slater. Academic Press, New York, pp 253–262
27. Alagona G, Tomasi J (1972) Theor Chem Acc 24:42
28. Santolini V, Malhado JP, Robb MA, Garavelli M, Bearpark MJ
(2015) Mol Phys 113:1978
29. Knowles PJ, Schütz M, Werner H-J (2000) In: Grotendorst J (ed)
Modern methods and algorithms of quantum chemistry, vol 3.
John von Neumann Institute for Computing, NIC Series, Jülich,
pp 97–179
30. Kirkwood JG (1932) Phys Z 33:57
31. Pople JA, Schofi eld P (1957) Philos Mag 2:591
32. Karplus M, Kolker HJ (1963a) J Chem Phys 39:1493
33. Karplus M, Kolker HJ (1963b) J Chem Phys 39:2997
34. Rivail J-L, Cartier A (1978) Mol Phys 36:1085
35. Rivail J-L, Cartier A (1979) Chem Phys Lett 61:469
36. Sadlej AJ, Jaszunski M (1971) Mol Phys 22:761
37. Toulouse J, Gerber IC, Jansen G, Savin A, Ángyán JG (2009)
Phys Rev Lett 102:096404
38. Zhu W, Toulouse J, Savin A, Ángyán JG (2010) J Chem Phys
132:244108
39. Toulouse J, Zhu W, Savin A, Jansen G, Ángyán JG (2011) J
Chem Phys 135:084119
40. Ángyán JG, Liu R-F, Toulouse J, Jansen G (2011) J Chem Theory Comput 7:3116
41. Kapuy E, Kozmutza C (1991) J Chem Phys 94:5565
42. Saebø S, Tong W, Pulay P (1993) J Chem Phys 98:2170
43. Hetzer G, Pulay P, Werner H-J (1998) Chem Phys Lett 290:143
44. Usvyat D, Maschio L, Manby FR, Casassa S, Schütz M, Pisani C
(2007) Phys Rev B 76:075102
45. Chermak E, Mussard B, Ángyán JG, Reinhardt P (2012) Chem
Phys Lett 550:162
46. Silvestrelli PL (2009) J Phys Chem A 113:5224
47. Silvestrelli PL, Benyahia K, Grubisic S, Ancilotto F, Toigo F
(2009) J Chem Phys 130:074702
48. Ambrosetti A, Silvestrelli PL (2012) Phys Rev B 85:073101
49. Silvestrelli PL (2013) J Chem Phys 139:054106
50. Silvestrelli PL, Ambrosetti A (2014) J Chem Phys 140:124107
51. Claverie P, Rein R (1969) Int J Quantum Chem 3:537
52. Claverie P (1978) In: Pullman B (ed) Intermolecular interactions: from diatomics to biopolymers, vol 1, chap. 2. Wiley, New
York, pp 69–305
53. Alcoba DR, Lain L, Torre A, Bochicchio RC (2006) J Comput
Chem 27:596
54. Pipek J, Mezey PG (1989) J Chem Phys 90:4916
55. Resta R (2006) J Chem Phys 124:104104
56. Ángyán JG, Gerber IC, Savin A, Toulouse J (2005) Phys Rev A
72:012510
57. Werner H-J, Knowles PJ, Knizia G, Manby FR, Schütz M (2012)
WIREs Comput Mol Sci 2:242–253. doi: 10.1002/wcms.82
58. Toulouse J, Rebolini E, Gould T, Dobson JF, Seal P, Ángyán JG
(2013) J Chem Phys 138:194106
59. Zeiss GD, Meath WJ, MacDonald J, Dawson DJ (1977) Can J
Phys 55:2080
113
Reprinted from the journal
1 3
In the present work, the exchange contribution, which
is present only in the case of Hartree–Fock or hybrid density functional fockians and which would give rise to nonconventional two-electron integrals, is not treated explicitly. Possible approximate solutions for this problem will
be discussed in forthcoming works. Although we do not
use the elements of the off-diagonal ( i = j ) blocks of the
POO fock operator, for the sake of completeness we give
its expression:
To derive this, instead of the double commutator of
Eq. 73 , one needs to consider the product of the commutator with a coordinate operator
where ˆ
T is the kinetic energy operator, and [ˆ r β , ˆ
T ] = ˆ
∇ β .
References
1. Perdew JP, Schmidt K (2001) In: van Doren V, van Alsenoy K,
Geerlings P (eds) AIP conference proceedings (AIP), pp 1–20
2. Eichkorn K, Treutler O, Ohm H, Haser M, Ahlrichs R (1995)
Chem Phys Lett 240:283
3. Whitten JL (1973) J Chem Phys 58:4496
4. Roeggen I, Johansen T (2008) J Chem Phys 128:194107
5. Beebe NHF, Linderberg J (1977) Int J Quantum Chem 12:683
6. Harl J, Kresse G (2008) Phys Rev B 77:045136
7. Nguyen H-V, de Gironcoli S (2009) Phys Rev B 79:205114
8. Wilson HF, Lu D, Gygi F, Galli G (2009) Phys Rev B 79:245106
9. Wilson HF, Gygi F, Galli G (2008) Phys Rev B 78:113303
10. Rocca D (2014) J Chem Phys 140:18A501
11. Surján PR (2005) Chem Phys Lett 406:318
12. Johnson ER, Becke AD (2005) J Chem Phys 123:024101
13. Johnson ER, Mackie ID, DiLabio GA (2009) J Phys Org Chem
22:1127
14. Grimme S, Antony J, Ehrlich S, Krieg H (2010) J Chem Phys
132:154104
15. Grimme S (2011) Wires Comput Mol Sci 1:211
16. Tkatchenko A, Scheffl er M (2009) Phys Rev Lett 102:073005
(81)
i α | ˆ
f |i β = 1
2 δ αβ − 1
2
occ
m
im|
ˆ
r α − ˆ
r
α
w
r, r
ˆ
r β − ˆ
r
β
|mi
+ 1
2
occ
m
i|ˆ r α ˆ
r β |mf mi + f im m|ˆ r α ˆ
r β |i
−
occ
mn
i|ˆ r α |mf mn n|ˆ r β |i.
(82)
i α | ˆ
f |j β = −−i |ˆ r α ˆ
∇ β |j +
occ
m
im|ˆ r α w
r, r
ˆ
r β − ˆ
r
β
|mj
+
occ
m
i |ˆ r α ˆ
r β |m f mi
−
occ
mn
i |ˆ r α |m f mn n|ˆ r β |i.
(83)
ˆ
r α
ˆ
r β , ˆ
f
= ˆ
r α
ˆ
r β , ˆ
T
− ˆ
r α
ˆ
r β , ˆ
K
,
17. DiStasio RA Jr, Gobre VV, Tkatchenko A (2014) J Phys Condens Matter 26:213202
18. Reilly AM, Tkatchenko A (2015) Chem Sci 6:3289
19. Surján PR (1989) In: Maksic ZB (ed) Theoretical models of
chemical bonding, vol 2. Springer, Heidelberg, pp 205–256
20. Høyvik I-M, Jansík B, Jørgensen P (2012) J Chem Phys
137:224114
21. Pulay P (1983) Chem Phys Lett 100:151
22. Pulay P, Saebø S (1986) Theor Chim Acta 69:357
23. Boughton JW, Pulay P (1993) J Comput Chem 14:736
24. Foster JM, Boys SF (1960a) Rev Mod Phys 32:300
25. Foster JM, Boys SF (1960b) Rev Mod Phys 32:303
26. Boys SF (1966) In: Löwdin PO (ed) Quantum theory of atoms,
molecules, and the solid state, a tribute to John C. Slater. Academic Press, New York, pp 253–262
27. Alagona G, Tomasi J (1972) Theor Chem Acc 24:42
28. Santolini V, Malhado JP, Robb MA, Garavelli M, Bearpark MJ
(2015) Mol Phys 113:1978
29. Knowles PJ, Schütz M, Werner H-J (2000) In: Grotendorst J (ed)
Modern methods and algorithms of quantum chemistry, vol 3.
John von Neumann Institute for Computing, NIC Series, Jülich,
pp 97–179
30. Kirkwood JG (1932) Phys Z 33:57
31. Pople JA, Schofi eld P (1957) Philos Mag 2:591
32. Karplus M, Kolker HJ (1963a) J Chem Phys 39:1493
33. Karplus M, Kolker HJ (1963b) J Chem Phys 39:2997
34. Rivail J-L, Cartier A (1978) Mol Phys 36:1085
35. Rivail J-L, Cartier A (1979) Chem Phys Lett 61:469
36. Sadlej AJ, Jaszunski M (1971) Mol Phys 22:761
37. Toulouse J, Gerber IC, Jansen G, Savin A, Ángyán JG (2009)
Phys Rev Lett 102:096404
38. Zhu W, Toulouse J, Savin A, Ángyán JG (2010) J Chem Phys
132:244108
39. Toulouse J, Zhu W, Savin A, Jansen G, Ángyán JG (2011) J
Chem Phys 135:084119
40. Ángyán JG, Liu R-F, Toulouse J, Jansen G (2011) J Chem Theory Comput 7:3116
41. Kapuy E, Kozmutza C (1991) J Chem Phys 94:5565
42. Saebø S, Tong W, Pulay P (1993) J Chem Phys 98:2170
43. Hetzer G, Pulay P, Werner H-J (1998) Chem Phys Lett 290:143
44. Usvyat D, Maschio L, Manby FR, Casassa S, Schütz M, Pisani C
(2007) Phys Rev B 76:075102
45. Chermak E, Mussard B, Ángyán JG, Reinhardt P (2012) Chem
Phys Lett 550:162
46. Silvestrelli PL (2009) J Phys Chem A 113:5224
47. Silvestrelli PL, Benyahia K, Grubisic S, Ancilotto F, Toigo F
(2009) J Chem Phys 130:074702
48. Ambrosetti A, Silvestrelli PL (2012) Phys Rev B 85:073101
49. Silvestrelli PL (2013) J Chem Phys 139:054106
50. Silvestrelli PL, Ambrosetti A (2014) J Chem Phys 140:124107
51. Claverie P, Rein R (1969) Int J Quantum Chem 3:537
52. Claverie P (1978) In: Pullman B (ed) Intermolecular interactions: from diatomics to biopolymers, vol 1, chap. 2. Wiley, New
York, pp 69–305
53. Alcoba DR, Lain L, Torre A, Bochicchio RC (2006) J Comput
Chem 27:596
54. Pipek J, Mezey PG (1989) J Chem Phys 90:4916
55. Resta R (2006) J Chem Phys 124:104104
56. Ángyán JG, Gerber IC, Savin A, Toulouse J (2005) Phys Rev A
72:012510
57. Werner H-J, Knowles PJ, Knizia G, Manby FR, Schütz M (2012)
WIREs Comput Mol Sci 2:242–253. doi: 10.1002/wcms.82
58. Toulouse J, Rebolini E, Gould T, Dobson JF, Seal P, Ángyán JG
(2013) J Chem Phys 138:194106
59. Zeiss GD, Meath WJ, MacDonald J, Dawson DJ (1977) Can J
Phys 55:2080
113
Reprinted from the journal