Theor Chem Acc (2015) 134:86
1 3
indices we can conclude that the two sums are equal. Thus
we have in the real case
The summation over ρ ∈ B; B = A in the right-hand side
of Eq. ( 51 ) means that ρ runs over all the orbital indices,
except those assigned to atom A ; we may add and subtract
the sum for the case ρ ∈ A :
In the fi rst term on the right-hand side, we can sum over
ρ to get the “projected density matrix element” B AC
τ η , while
in the second term the coeffi cient A AC
ρη again reduces to the
Kronecker delta δ ρη . Thus we get, changing the summation
index C to B in the right-hand side:
(51)
B,C
B,C =A
ρ∈B
τ ∈C
D τρ [μρ|ντ ]
=⇒
B,C
B,C =A
ρ∈B
τ ∈C
D τρ
η∈AC
A
AC
ρη [μη|ντ ].
(52)
B,C
B,C =A
ρ∈B
τ ∈C
D τρ [μρ|ντ ] =⇒
C
C =A
ρ
τ ∈C
D τρ
η∈AC
A
AC
ρη [μη|ντ ]
−
C
C =A
ρ∈A
τ ∈C
D τρ
η∈AC
A
AC
ρη [μη|ντ ].
(53)
B,C
B,C =A
ρ∈B
τ ∈C
D τρ [μρ|ντ ] =⇒
B
B =A
τ ∈B
η∈AB
B
AB
τ η [μη|ντ ]
−
B
B =A
ρ∈A
τ ∈B
D τρ [μρ|ντ ].
The second sum just cancels the respective term in the third
line of Eq. ( 48 ).
The second (exchange) term in the fourth line of Eq. ( 48 )
transforms analogously. However, in that case the two
terms are not equal, as were in Eq. ( 50 ), and there is no full
canceling of the second term in the third line of Eq. ( 48 );
instead the half of the respective terms in both second and
third lines is canceled.
References
1. Mayer I (1983) Int J Quantum Chem 23:341
2. Ruedenberg K (1951) J Chem Phys 19:1433
3. Mayer I (1998) Int J Quantum Chem 70:41
4. Salvador P, Asturiol D, Mayer I (2006) J Comput Chem 27:1505
5. Mayer I (1983) Chem Phys Lett 97:270
6. Mayer I (2007) J Comput Chem 28:204
7. Mayer I (2012) Chem Phys Lett 544:83
8. Programs BORDER, NEWBORDER etc. http://occam.ttk.mta.
hu
9. Mayer I (2000) Chem Phys Lett 332:381
10. Program APOST. http://occam.ttk.mta.hu
11. Mayer I (2006) Phys Chem Chem Phys 8:4630
12. Mayer I (2007) Faraday Discuss 135:439
13. Mayer I (2012) Phys Chem Chem Phys 14:337
14. Programs APEX4, ENPART, NEWENPART. http://occam.ttk.
mta.hu
15. Maseras F, Morokuma K (1995) J Comput Chem 16:1170
16. Longuet-Higgins HC (1966) In: Löwdin P-O (ed) Quantum
theory of atoms, molecules and the solid state. Academic Press,
New York, p 105
17. Surján PR (1989) Second quantized approach to quantum chemistry. Springer, Berlin
18. Hamza A, Mayer I (2003) Theor Chem Acc 109:91
40
Reprinted from the journal
1 3
indices we can conclude that the two sums are equal. Thus
we have in the real case
The summation over ρ ∈ B; B = A in the right-hand side
of Eq. ( 51 ) means that ρ runs over all the orbital indices,
except those assigned to atom A ; we may add and subtract
the sum for the case ρ ∈ A :
In the fi rst term on the right-hand side, we can sum over
ρ to get the “projected density matrix element” B AC
τ η , while
in the second term the coeffi cient A AC
ρη again reduces to the
Kronecker delta δ ρη . Thus we get, changing the summation
index C to B in the right-hand side:
(51)
B,C
B,C =A
ρ∈B
τ ∈C
D τρ [μρ|ντ ]
=⇒
B,C
B,C =A
ρ∈B
τ ∈C
D τρ
η∈AC
A
AC
ρη [μη|ντ ].
(52)
B,C
B,C =A
ρ∈B
τ ∈C
D τρ [μρ|ντ ] =⇒
C
C =A
ρ
τ ∈C
D τρ
η∈AC
A
AC
ρη [μη|ντ ]
−
C
C =A
ρ∈A
τ ∈C
D τρ
η∈AC
A
AC
ρη [μη|ντ ].
(53)
B,C
B,C =A
ρ∈B
τ ∈C
D τρ [μρ|ντ ] =⇒
B
B =A
τ ∈B
η∈AB
B
AB
τ η [μη|ντ ]
−
B
B =A
ρ∈A
τ ∈B
D τρ [μρ|ντ ].
The second sum just cancels the respective term in the third
line of Eq. ( 48 ).
The second (exchange) term in the fourth line of Eq. ( 48 )
transforms analogously. However, in that case the two
terms are not equal, as were in Eq. ( 50 ), and there is no full
canceling of the second term in the third line of Eq. ( 48 );
instead the half of the respective terms in both second and
third lines is canceled.
References
1. Mayer I (1983) Int J Quantum Chem 23:341
2. Ruedenberg K (1951) J Chem Phys 19:1433
3. Mayer I (1998) Int J Quantum Chem 70:41
4. Salvador P, Asturiol D, Mayer I (2006) J Comput Chem 27:1505
5. Mayer I (1983) Chem Phys Lett 97:270
6. Mayer I (2007) J Comput Chem 28:204
7. Mayer I (2012) Chem Phys Lett 544:83
8. Programs BORDER, NEWBORDER etc. http://occam.ttk.mta.
hu
9. Mayer I (2000) Chem Phys Lett 332:381
10. Program APOST. http://occam.ttk.mta.hu
11. Mayer I (2006) Phys Chem Chem Phys 8:4630
12. Mayer I (2007) Faraday Discuss 135:439
13. Mayer I (2012) Phys Chem Chem Phys 14:337
14. Programs APEX4, ENPART, NEWENPART. http://occam.ttk.
mta.hu
15. Maseras F, Morokuma K (1995) J Comput Chem 16:1170
16. Longuet-Higgins HC (1966) In: Löwdin P-O (ed) Quantum
theory of atoms, molecules and the solid state. Academic Press,
New York, p 105
17. Surján PR (1989) Second quantized approach to quantum chemistry. Springer, Berlin
18. Hamza A, Mayer I (2003) Theor Chem Acc 109:91
40
Reprinted from the journal