$
t
$E eff ¼ $
t
$G À q$
t
$V mep À $
t V mep $q À V mep $
t
$q
À $
t
q$V mep À $
t
$V xc :
ð6:41Þ
where $
t
$V xc r
ð Þ is given by Eq. 6.34 The gradient and Hessian of E add r
ð Þ,
Eq. 6.20, are even more complicated:
$E add ¼ $G À q$V
nuc
mep À V
nuc
mep $q
À
1
2
q$V
ele
mep þ V
ele
mep $q þ F xc
h
i
À
1
2
X
i
g i V G i $G i ;
ð6:42Þ
$
t
$E add ¼ $
t
$G À q$
t
$V
nuc
mep À $
t V
nuc
mep $q À V
nuc
mep $
t
$q À $
t
q$V
nuc
mep
À
1
2
H xc þ q$
t
$V
ele
mep þ $
t V
ele
mep $q þ V
ele
mep $
t
$q þ $
t
q$V
ele
mep
h
i
À
1
2
X
i
g i $
t G i $V G i þ V G i $
t
$G i
½
:
ð6:43Þ
To shorten Eqs. 6.39–6.43, the dependence on r of every magnitude has been
avoided.
References
1. Bader RFW (1990) Atoms in molecules. Oxford University Press, Oxford
2. Bader RFW, Gatti C (1998) Chem Phys Lett 287:233–238
3. Becke AD, Edgecombe KE (1990) A simple measure of electron localization in atomic and
molecular systems. J Chem Phys 92:5397–5403
4. Berlin T (1951) J Chem Phys 19:208–213
5. Blanco MA, Martín Pendás A, Francisco E (2005) J Chem Theory Comput 1:1096–1109
6. Cancès E, Keriven R, Lodier F, Savin A (2004) Theoret Chem Acc 111:373–380
7. Dong-Xia Z, Li-Dong G, Zhong-Zhi Y (2005) J Phys Chem 109:10121–10128
8. Dong-Xia Z, Zhong-Zhi Y (2014) J Comput Chem 35:965–977
9. Finzel K (2014) Int J Quantum Chem 114(9):568–576
10. Francisco E, Martín Pendás A, Blanco MA J Chem Phys 126
11. Francisco E, Martín Pendás A, Blanco MA (2008) Comput Phys Commun 178:621–634
12. Francisco E, Martín Pendás A, García-Revilla M (2013) Comput Theoret Chem 1003:71–78
13. Francisco E, Pendás M (2014) Comput Phys Commun 185:2663–2682
14. García-Revilla M, Francisco E, Popelier PLA, Martín Pendás A (2013) Chem Phys Chem
14:1211–1218
15. García-Revilla M, Popelier PLA, Francisco E, Martín Pendás A (2011) J Chem Theory
Comput 7:1704–1711
16. Gatti C, Macchi P (eds) (2012) Modern charge-density analysis. Springer, Dordrecht
17. Guevara-Vela JM, Chávez-Calvillo R, García-Revilla M, Hernández-Trujillo J,
Christiansen O, Francisco E, Martín Pendás A (2013) Chem Eur J 19, 14:304–14,315
18. Hernandez-Trujillo J, Cortés-Guzman F, Fang DC, Bader RFW (2007) Faraday Discuss
135:79–95
19. Kohout M (2004) A measure of electron localizability. Int J Quantum Chem 97:651–658
6 Emergent Scalar and Vector Fields in Quantum Chemical Topology
149
t
$E eff ¼ $
t
$G À q$
t
$V mep À $
t V mep $q À V mep $
t
$q
À $
t
q$V mep À $
t
$V xc :
ð6:41Þ
where $
t
$V xc r
ð Þ is given by Eq. 6.34 The gradient and Hessian of E add r
ð Þ,
Eq. 6.20, are even more complicated:
$E add ¼ $G À q$V
nuc
mep À V
nuc
mep $q
À
1
2
q$V
ele
mep þ V
ele
mep $q þ F xc
h
i
À
1
2
X
i
g i V G i $G i ;
ð6:42Þ
$
t
$E add ¼ $
t
$G À q$
t
$V
nuc
mep À $
t V
nuc
mep $q À V
nuc
mep $
t
$q À $
t
q$V
nuc
mep
À
1
2
H xc þ q$
t
$V
ele
mep þ $
t V
ele
mep $q þ V
ele
mep $
t
$q þ $
t
q$V
ele
mep
h
i
À
1
2
X
i
g i $
t G i $V G i þ V G i $
t
$G i
½
:
ð6:43Þ
To shorten Eqs. 6.39–6.43, the dependence on r of every magnitude has been
avoided.
References
1. Bader RFW (1990) Atoms in molecules. Oxford University Press, Oxford
2. Bader RFW, Gatti C (1998) Chem Phys Lett 287:233–238
3. Becke AD, Edgecombe KE (1990) A simple measure of electron localization in atomic and
molecular systems. J Chem Phys 92:5397–5403
4. Berlin T (1951) J Chem Phys 19:208–213
5. Blanco MA, Martín Pendás A, Francisco E (2005) J Chem Theory Comput 1:1096–1109
6. Cancès E, Keriven R, Lodier F, Savin A (2004) Theoret Chem Acc 111:373–380
7. Dong-Xia Z, Li-Dong G, Zhong-Zhi Y (2005) J Phys Chem 109:10121–10128
8. Dong-Xia Z, Zhong-Zhi Y (2014) J Comput Chem 35:965–977
9. Finzel K (2014) Int J Quantum Chem 114(9):568–576
10. Francisco E, Martín Pendás A, Blanco MA J Chem Phys 126
11. Francisco E, Martín Pendás A, Blanco MA (2008) Comput Phys Commun 178:621–634
12. Francisco E, Martín Pendás A, García-Revilla M (2013) Comput Theoret Chem 1003:71–78
13. Francisco E, Pendás M (2014) Comput Phys Commun 185:2663–2682
14. García-Revilla M, Francisco E, Popelier PLA, Martín Pendás A (2013) Chem Phys Chem
14:1211–1218
15. García-Revilla M, Popelier PLA, Francisco E, Martín Pendás A (2011) J Chem Theory
Comput 7:1704–1711
16. Gatti C, Macchi P (eds) (2012) Modern charge-density analysis. Springer, Dordrecht
17. Guevara-Vela JM, Chávez-Calvillo R, García-Revilla M, Hernández-Trujillo J,
Christiansen O, Francisco E, Martín Pendás A (2013) Chem Eur J 19, 14:304–14,315
18. Hernandez-Trujillo J, Cortés-Guzman F, Fang DC, Bader RFW (2007) Faraday Discuss
135:79–95
19. Kohout M (2004) A measure of electron localizability. Int J Quantum Chem 97:651–658
6 Emergent Scalar and Vector Fields in Quantum Chemical Topology
149