Using an approach based upon a set of auxiliary many-electron wavefunctions we present a rigorous derivation of spin-orbit coupling (SOC) within the framework of linear-response time-dependent density functional theory (LR-TDDFT). Our method is based on a perturbative correction of the non-relativistic collinear TDDFT equations using a Breit-Pauli spin-orbit Hamiltonian. The derivation, which is performed within both the Casida and Sternheimer formulations of LR-TDDFT, is valid for any basis set. The requirement of spin noncollinearity for the treatment of spin-flip transitions is also discussed and a possible alternative solution for the description of these transitions in the collinear case is also proposed. Our results are validated by computing the SOC matrix elements between singlet and triplet states of two molecules, formaldehyde and acetone. In both cases, we find excellent agreement with benchmark calculations performed with a high level correlated wavefunction method.

3.
J.
Autschbach
,
J. Chem. Phys.
136
,
150902
(
2012
).
4.
G.
Tarczay
,
A. G.
Császár
,
W.
Klopper
, and
H. M.
Quiney
,
Mol. Phys.
99
,
1769
(
2001
).
5.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
140
,
A1133
(
1965
).
6.
E.
Runge
and
E. K. U.
Gross
,
Phys. Rev. Lett.
52
,
997
(
1984
).
8.
A.
MacDonald
and
S. H.
Vosko
,
J. Phys. C
12
,
2997
(
1979
).
9.
F.
Wang
,
T.
Ziegler
,
E.
van Lenthe
,
S.
van Gisbergen
, and
E. J.
Baerends
,
J. Chem. Phys.
122
,
204103
(
2005
).
10.
F.
Wang
and
T.
Ziegler
,
J. Chem. Phys.
123
,
154102
(
2005
).
11.
I.
Tavernelli
,
E.
Tapavicza
, and
U.
Rothlisberger
,
J. Chem. Phys.
130
,
124107
(
2009
).
12.
I.
Tavernelli
,
E.
Tapavicza
, and
U.
Rothlisberger
,
J. Mol. Struct.: THEOCHEM
914
,
22
(
2009
).
13.
I.
Tavernelli
,
B. F. E.
Curchod
, and
U.
Rothlisberger
,
J. Chem. Phys.
131
,
196101
(
2009
).
14.
I.
Tavernelli
,
B. F. E.
Curchod
,
A.
Laktionov
, and
U.
Rothlisberger
,
J. Chem. Phys.
133
,
194104
(
2010
).
15.
CPMD, copyright IBM Corp 1990-2008, Copyright MPI für Festkörperforschung Stuttgart 1997-2001, see http://www.cpmd.org/.
16.
M. E.
Casida
in
Recent Advances in Density Functional Methods
, edited by
D. P.
Chong
(
World Scientific
,
Singapore
,
1995
), p.
155
.
17.
S. G.
Chiodo
and
N.
Russo
,
Chem. Phys. Lett.
490
,
90
(
2010
).
18.
Q.
Ou
and
J. E.
Subotnik
,
J. Phys. Chem. C
117
,
19839
(
2013
).
19.
R.
Sternheimer
,
Phys. Rev.
84
,
244
(
1951
).
20.
J.
Hutter
,
J. Chem. Phys.
118
,
3928
(
2003
).
21.
G.
Granucci
,
M.
Persico
, and
G.
Spighi
,
J. Chem. Phys.
137
,
22A501
(
2012
).
22.
P.
Marquetand
,
M.
Richter
,
J.
González-Vázquez
,
I.
Sola
, and
L.
González
,
Faraday Discuss.
153
,
261
(
2011
).
23.
A. K.
Rajagopal
and
J.
Callaway
,
Phys. Rev. B
7
,
1912
(
1973
).
24.
R. M.
Dreizler
and
E. K. U.
Gross
,
Density Functional Theory
(
Springer
,
Berlin
,
1990
).
25.
U.
von Barth
and
L.
Hedin
,
J. Phys. C: Solid State Phys.
5
,
1629
(
1972
).
26.
O.
Gunnarsson
and
B. I.
Lundqvist
,
Phys. Rev. B
13
,
4274
(
1976
).
27.
C. A.
Ullrich
,
Time-Dependent Density-Functional Theory
(
Oxford University Press
,
2012
).
28.
K. L.
Liu
and
S. H.
Vosko
,
Can. J. Phys.
67
,
1015
(
1989
).
29.
Y.
Shao
,
M.
Head-Gordon
, and
A. I.
Krylov
,
J. Chem. Phys.
118
,
4807
(
2003
).
30.
F.
Wang
and
T.
Ziegler
,
J. Chem. Phys.
121
,
12191
(
2004
).
31.
F.
Wang
and
T.
Ziegler
,
J. Chem. Phys.
122
,
074109
(
2005
).
32.
Time-Dependent Density Functional Theory
,
Lecture Notes in Physics
, edited by
M. A. L.
Marques
,
C. A.
Ullrich
,
F.
Nogueira
,
A.
Rubio
,
K.
Burke
, and
E. K. U.
Gross
(
Springer
,
Berlin
,
2006
).
33.
E.
Tapavicza
,
I.
Tavernelli
, and
U.
Rothlisberger
,
Phys. Rev. Lett.
98
,
023001
(
2007
).
34.
M. A. L.
Marques
and
E. K. U.
Gross
,
Lect. Notes Phys.
620
,
144
(
2003
).
35.
The definition of the TDDFT kernel K(ω) already incorporates the adiabatic approximation, in which
$\delta v_{xc}^{\sigma }({\bf r},t)/{\delta \rho _{\tau }(\mathbf {r^{\prime }},t^{\prime })}={\scriptstyle \delta (t-t^{\prime })}\delta ^2 E_{xc}/\break {(\delta \rho _{\sigma }(\mathbf {r},t) \delta \rho _{\tau }(\mathbf {r^{\prime }},t^{\prime })) }$
δvxcσ(r,t)/δρτ(r,t)=δ(tt)δ2Exc/(δρσ(r,t)δρτ(r,t))
, where vxc is the exchange-correlation potential.
36.
X.
Gonze
and
J.-P.
Vigneron
,
Phys. Rev. B
39
,
13120
(
1989
).
38.
X.
Andrade
,
S.
Botti
,
M. A. L.
Marques
, and
A.
Rubio
,
J. Chem. Phys.
126
,
184106
(
2007
).
39.
R.
McWeeny
,
Methods of Molecular Quantum Mechanics
(
Academic Press
,
1992
).
40.
P.-O.
Löwdin
,
Phys. Rev.
97
,
1474
(
1955
).
41.
B. A.
Hess
,
C. M.
Marian
,
U.
Wahgren
, and
O.
Groppen
,
Chem. Phys. Lett.
251
,
365
(
1996
).
42.
D.
Sebastiani
and
M.
Parrinello
,
J. Chem. Phys.
105
,
1951
(
2001
).
43.
M. A.
El-Sayed
,
J. Chem. Phys.
38
,
2834
(
1963
).
44.
M. W.
Schmidt
,
K. K.
Baldridge
,
J. A.
Boatz
,
S. T.
Elber
,
M. S.
Gordon
,
J. H.
Jensen
,
S.
Koseki
,
N.
Matsunaga
,
K. A.
Nguyen
,
S. J.
Su
 et al.,
J. Comput. Chem.
14
,
1347
1363
(
1993
).
45.
Perturbation theory calculation based on a wavefunction obtained from a complete-active space multi-configuration self-consistent field calculation (CASSCF).
46.
ADF2009.01, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, see http://www.scm.com,
2009
.
47.
N.
Troullier
and
J. L.
Martins
,
Phys. Rev. B
43
,
8861
(
1991
).
48.
S.
Koseki
,
M. S.
Gordon
,
M. W.
Schmidt
, and
N.
Matsunga
,
J. Phys. Chem.
99
,
12764
(
1995
).
49.
C. M.
Marian
, “
Berechnung von Matrixelementen des Spin-Bhan- und Spin-Spin-Kopplungsoperators mit MRD-CI-Wellenfunktionen
,” Ph.D. thesis (
University of Bonn
,
1981
).
50.
Y. S.
Lee
,
W. C.
Ermler
, and
K. S.
Pitzer
,
J. Phys. Chem.
67
,
5861
(
1977
).
51.
P.
Hafner
and
W. H.
Schwarz
,
J. Phys. B
11
,
217
(
1978
).
52.
C. H.
Teichteil
,
M.
Pĺissier
, and
F.
Speigelmann
,
Chem. Phys.
81
,
273
(
1983
).
53.
J. C.
Tully
and
R. K.
Preston
,
J. Chem. Phys.
55
,
562
(
1971
).
54.
L.
Martínez-Fernández
,
I.
Corral
,
G.
Granucci
, and
M.
Persico
,
Chem. Sci.
5
,
1336
(
2014
).
55.
M.
Richter
,
P.
Marquetand
,
J.
González-Vázquez
,
I.
Sola
, and
L.
González
,
J. Chem. Theory Comput.
7
,
1253
(
2011
).
56.
B. F. E.
Curchod
,
I.
Tavernelli
, and
U.
Rothlisberger
,
Phys. Chem. Chem. Phys.
13
,
3231
(
2011
).
57.
B. F. E.
Curchod
and
I.
Tavernelli
,
J. Chem. Phys.
138
,
184112
(
2013
).
58.
B. F. E.
Curchod
,
U.
Rothlisberger
, and
I.
Tavernelli
,
ChemPhysChem
14
,
1314
(
2013
).
59.
N.
Troullier
and
J. L.
Martins
,
Phys. Rev. B
43
,
1993
(
1991
).
60.
I.
Tamm
,
J. Phys. (Moscow)
9
,
449
(
1945
).
61.
62.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
63.
T. H. J.
Dunning
,
J. Chem. Phys.
90
,
1007
(
1989
).
64.
H.
Nakano
,
J. Chem. Phys.
99
,
7983
(
1993
).
65.
D. G.
Fedorov
and
J. P.
Finley
,
Phys. Rev. A
64
,
042502
(
2001
).
You do not currently have access to this content.