The fully correlated frequency-independent Dirac–Coulomb–Breit Hamiltonian provides the most accurate description of electron–electron interaction before going to a genuine relativistic quantum electrodynamics theory of many-electron systems. In this work, we introduce a correlated Dirac–Coulomb–Breit multiconfigurational self-consistent-field method within the frameworks of complete active space and density matrix renormalization group. In this approach, the Dirac–Coulomb–Breit Hamiltonian is included variationally in both the mean-field and correlated electron treatment. We also analyze the importance of the Breit operator in electron correlation and the rotation between the positive- and negative-orbital space in the no-virtual-pair approximation. Atomic fine-structure splittings and lanthanide contraction in diatomic fluorides are used as benchmark studies to understand the contribution from the Breit correlation.
REFERENCES
1.
J. B.
Mann
and W. R.
Johnson
, “Breit interaction in multielectron atoms
,” Phys. Rev. A
4
, 41
–51
(1971
).2.
K.-N.
Huang
, M.
Aoyagi
, M. H.
Chen
, B.
Crasemann
, and H.
Mark
, “Neutral-atom electron binding energies from relaxed-orbital relativistic Hartree-Fock-slater calculations 2 ≤ Z ≤ 106
,” At. Data Nucl. Data Tables
18
, 243
–291
(1976
).3.
C.
Chantler
, T.
Nguyen
, J.
Lowe
, and I.
Grant
, “Convergence of the Breit interaction in self-consistent and configuration-interaction approaches
,” Phys. Rev. A
90
, 062504
(2014
).4.
K.
Kozioł
, C. A.
Giménez
, and G. A.
Aucar
, “Breit corrections to individual atomic and molecular orbital energies
,” J. Chem. Phys.
148
, 044113
(2018
).5.
S.
Sun
, J.
Ehrman
, Q.
Sun
, and X.
Li
, “Efficient evaluation of the Breit operator in the Pauli spinor basis
,” J. Chem. Phys.
157
, 064112
(2022
).6.
C.
Thierfelder
and P.
Schwerdtfeger
, “Quantum electrodynamic corrections for the valence shell in heavy many-electron atoms
,” Phys. Rev. A
82
, 062503
(2010
).7.
P.
Pyykkö
, “The physics behind chemistry and the periodic table
,” Chem. Rev.
112
, 371
–384
(2012
).8.
P.
Pyykkö
, “Relativistic effects in chemistry: More common than you thought
,” Annu. Rev. Phys. Chem.
63
, 45
–64
(2012
).9.
W.
Liu
and I.
Lindgren
, “Going beyond ‘No-pair relativistic quantum chemistry
,’” J. Chem. Phys.
139
, 014108
(2013
).10.
W.
Liu
, “Advances in relativistic molecular quantum mechanics
,” Phys. Rep.
537
, 59
–89
(2014
).11.
W.
Liu
, “Essentials of relativistic quantum chemistry
,” J. Chem. Phys.
152
, 180901
(2020
).12.
I. P.
Grant
and W. G.
Penney
, “Relativistic self-consistent fields
,” Proc. R. Soc. London, Ser. A
262
, 555
–576
(1961
).13.
E.
Lindroth
, A.-M.
Martensson-Pendrill
, A.
Ynnerman
, and P.
Oster
, “Self-consistent treatment of the Breit interaction, with application to the electric dipole moment in thallium
,” J. Phys. B
22
, 2447
–2464
(1989
).14.
L.
Visscher
, T. J.
Lee
, and K. G.
Dyall
, “Formulation and implementation of a relativistic unrestricted coupled-cluster method including noniterative connected triples
,” J. Chem. Phys.
105
, 8769
(1996
).15.
J.
Thyssen
, T.
Fleig
, and H. J. A.
Jensen
, “A direct relativistic four-component multiconfiguration self-consistent-field method for molecules
,” J. Chem. Phys.
129
, 034109
(2008
).16.
J. E.
Bates
and T.
Shiozaki
, “Fully relativistic complete active space self-consistent field for large molecules: Quasi-second-order minimax optimization
,” J. Chem. Phys.
142
, 044112
(2015
).17.
J. P.
Desclaux
, “Multiconfiguration relativistic DIRAC-FOCK program
,” Comput. Phys. Commun.
9
, 31
–45
(1975
).18.
I. P.
Grant
and B. J.
McKenzie
, “The transverse electron-electron interaction in atomic structure calculations
,” J. Phys. B: At. Mol. Phys.
13
, 2671
(1980
).19.
K. G.
Dyall
, I. P.
Grant
, C. T.
Johnson
, F. A.
Parpia
, and E. P.
Plummer
, “GRASP: A general-purpose relativistic atomic structure program
,” Comput. Phys. Commun.
55
, 425
–456
(1989
).20.
H. M.
Quiney
, I. P.
Grant
, and S.
Wilson
, “The Dirac equation in the algebraic approximation. V. Self-consistent field studies including the Breit interaction
,” J. Phys. B: At. Mol. Phys.
20
, 1413
–1422
(1987
).21.
H. M.
Quiney
, H.
Skaane
, and I. P.
Grant
, “Relativistic calculation of electromagnetic interactions in molecules
,” J. Phys. B: At. Mol. Phys.
30
, L829
–L834
(1997
).22.
H. M.
Quiney
, H.
Skaane
, and I. P.
Grant
, “Relativistic, quantum electrodynamic and many-body effects in the water molecule
,” Chem. Phys. Lett.
290
, 473
–480
(1998
).23.
Y.
Ishikawa
, “Dirac–Fock Gaussian basis calculations: Inclusion of the Breit interaction in the self-consistent field procedure
,” Chem. Phys. Lett.
166
, 321
–325
(1990
).24.
Y.
Ishikawa
, H. M.
Quiney
, and G. L.
Malli
, “Dirac-Fock-Breit self-consistent-field method: Gaussian basis-set calculations on many-electron atoms
,” Phys. Rev. A
43
, 3270
–3278
(1991
).25.
F. A.
Parpia
, A. K.
Mohanty
, and E.
Clementi
, “Relativistic calculations for atoms: Self-consistent treatment of Breit interaction and nuclear volume effect
,” J. Phys. B: At. Mol. Phys.
25
, 1
–16
(1992
).26.
T.
Shiozaki
, “Communication: An efficient algorithm for evaluating the Breit and spin-spin coupling integrals
,” J. Chem. Phys.
138
, 111101
(2013
).27.
M. S.
Kelley
and T.
Shiozaki
, “Large-scale Dirac–Fock–Breit method using density fitting and 2-spinor basis functions
,” J. Chem. Phys.
138
, 204113
(2013
).28.
R. D.
Reynolds
, T.
Yanai
, and T.
Shiozaki
, “Large-scale relativistic complete active space self-consistent field with robust convergence
,” J. Chem. Phys.
149
, 014106
(2018
).29.
S.
Sun
, T. F.
Stetina
, T.
Zhang
, H.
Hu
, E. F.
Valeev
, Q.
Sun
, and X.
Li
, “Efficient four-component Dirac–Coulomb–Gaunt Hartree–Fock in the Pauli spinor representation
,” J. Chem. Theory Comput.
17
, 3388
–3402
(2021
).30.
J. M.
Kasper
, A. J.
Jenkins
, S.
Sun
, and X.
Li
, “Perspective on Kramers symmetry breaking and restoration in relativistic electronic structure methods for open-shell systems
,” J. Chem. Phys.
153
, 090903
(2020
).31.
C. E.
Hoyer
, H.
Hu
, L.
Lu
, S.
Knecht
, and X.
Li
, “Relativistic Kramers-unrestricted exact-two-component density matrix renormalization group
,” J. Phys. Chem. A
126
, 5011
–5020
(2022
).32.
R. E.
Stanton
and S.
Havriliak
, “Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations
,” J. Chem. Phys.
81
, 1910
–1918
(1984
).33.
Y.
Ishikawa
, R. C.
Binning
, and K. M.
Sando
, “Dirac-Fock discrete-basis calculations on the beryllium atom
,” Chem. Phys. Lett.
101
, 111
–114
(1983
).34.
K. G.
Dyall
and K.
Fægri
, “Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set
,” Chem. Phys. Lett.
174
, 25
–32
(1990
).35.
Q.
Sun
, W.
Liu
, and W.
Kutzelnigg
, “Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations
,” Theor. Chem. Acc.
129
, 423
–436
(2011
).36.
W.
Liu
, “Ideas of relativistic quantum chemistry
,” Mol. Phys.
108
, 1679
–1706
(2010
).37.
S.
Sun
, T. F.
Stetina
, T.
Zhang
, and X.
Li
, “On the finite nuclear effect and gaussian basis set for four-component Dirac Hartree-Fock calculations
,” in Rare Earth Elements and Actinides: Progress in Computational Science Applications
, edited by D. A.
Penchoff
, T. L.
Windus
, and C. C.
Peterson
(American Chemical Society
, 2021
), Vol. 1388, pp. 207
–218
.38.
J.
Sucher
, “Foundations of the relativistic theory of many-electron atoms
,” Phys. Rev. A
22
, 348
(1980
).39.
T.
Saue
and L.
Visscher
, in Theoretical Chemistry and Physics of Heavy and Superheavy Elements
, edited by S.
Wilson
and U.
Kaldor
(Kluwer
, Dordrecht
, 2003
), pp. 211
–261
.40.
A.
Almoukhalalati
, S.
Knecht
, H. J. A.
Jensen
, K. G.
Dyall
, and T.
Saue
, “Electron correlation within the relativistic no-pair approximation
,” J. Chem. Phys.
145
, 074104
(2016
).41.
P. J.
Knowles
and N. C.
Handy
, “A new determinant-based full configuration interaction method
,” Chem. Phys. Lett.
111
, 315
–321
(1984
).42.
S. R.
White
, “Density-matrix algorithms for quantum renormalization groups
,” Phys. Rev. B
48
, 10345
(1993
).43.
S.
Knecht
, Ö.
Legeza
, and M.
Reiher
, “Communication: Four-component density matrix renormalization group
,” J. Chem. Phys.
140
, 041101
(2014
).44.
S.
Battaglia
, S.
Keller
, and S.
Knecht
, “Efficient relativistic density-matrix renormalization group implementation in a matrix-product formulation
,” J. Chem. Theory Comput.
14
, 2353
–2369
(2018
).45.
H.
Zhai
and G. K.-L.
Chan
, “A comparison between the one- and two-step spin–orbit coupling approaches based on the ab initio density matrix renormalization group
,” J. Chem. Phys.
157
, 164108
(2022
).46.
D. B.
Williams-Young
, A.
Petrone
, S.
Sun
, T. F.
Stetina
, P.
Lestrange
, C. E.
Hoyer
, D. R.
Nascimento
, L.
Koulias
, A.
Wildman
, J.
Kasper
et al., “The Chronus Quantum (ChronusQ) software package
,” WIREs Comput. Mol. Sci.
10
, e1436
(2020
).47.
S.
Keller
, M.
Dolfi
, M.
Troyer
, and M.
Reiher
, “An efficient matrix product operator representation of the quantum chemical Hamiltonian
,” J. Chem. Phys.
143
, 244118
(2015
).48.
S.
Keller
and M.
Reiher
, “Spin-adapted matrix product states and operators
,” J. Chem. Phys.
144
, 134101
(2016
).49.
S.
Knecht
, E. D.
Hedegård
, S.
Keller
, A.
Kovyrshin
, Y.
Ma
, A.
Muolo
, C. J.
Stein
, and M.
Reiher
, “New approaches for ab initio calculations of molecules with strong electron correlation
,” Chimia
70
, 244
–251
(2016
).50.
I. P.
McCulloch
, “From density-matrix renormalization group to matrix product states
,” J. Stat. Mech.
2007
, P10014
.51.
H. J.
Werner
and W.
Meyer
, “A quadratically convergent MCSCF method for the simultaneous optimization of several states a quadratically convergent MCSCF method for the simultaneous optimization of several states
,” J. Chem. Phys.
74
, 5794
(1981
).52.
S.
Wouters
and D.
Van Neck
, “The density matrix renormalization group for ab initio quantum chemistry
,” Eur. Phys. J. D
68
, 1
–20
(2014
).53.
T.
Helgaker
, P.
Jørgensen
, and J.
Olsen
, Molecular Electronic-Structure Theory
, 1st ed. (John Wiley & Sons
, West Sussex, UK
, 2000
).54.
H.
Hu
, A. J.
Jenkins
, H.
Liu
, J. M.
Kasper
, M. J.
Frisch
, and X.
Li
, “Relativistic two-component multireference configuration interaction method with tunable correlation space
,” J. Chem. Theory Comput.
16
, 2975
–2984
(2020
).55.
A.
Kramida
, Y.
Ralchenko
, J.
Reader
NIST ASD Team
, NIST Atomic Spectra Database (version 5.6.1). https://physics.nist.gov/asd, 2018
, National Institute of Standards and Technology
, Gaithersburg, MD
.56.
B. O.
Roos
, R.
Lindh
, P.-Å.
Malmqvist
, V.
Veryazov
, and P.-O.
Widmark
, “Main group atoms and dimers studied with a new relativistic ANO basis set
,” J. Phys. Chem. A
108
, 2851
–2858
(2004
).57.
B. O.
Roos
, R.
Lindh
, P.-Å.
Malmqvist
, V.
Veryazov
, P.-O.
Widmark
, and A. C.
Borin
, “New relativistic atomic natural orbital basis sets for lanthanide atoms with applications to the Ce diatom and LuF3
,” J. Phys. Chem. A
112
, 11431
–11435
(2008
).58.
L. A.
Kaledin
, A. L.
Kaledin
, and M. C.
Heaven
, “Laser absorption spectroscopy of LaF: Analysis of the B1Π − X1Σ+ transition
,” J. Mol. Spectrosc.
182
, 50
–56
(1997
).59.
K. P.
Huber
and G.
Herzberg
, Molecular Spectra and Molecular Structure: IV. Constants of Diatomic Molecules
(Van Nostrand
, New York
, 1979
).60.
P. S.
Bagus
, Y. S.
Lee
, and K. S.
Pitzer
, “Effects of relativity and of the lanthanide contraction on the atoms from hafnium to bismuth
,” Chem. Phys. Lett.
33
, 408
–411
(1975
).61.
P.
Pyykkö
, “Dirac-Fock one-centre calculations Part 8. The 1Σ states of ScH,YH, LaH, AcH, TmH, LuH and LrH
,” Phys. Scr.
20
, 647
–651
(1979
).62.
W.
Küchle
, M.
Dolg
, and H.
Stoll
, “Ab initio study of the lanthanide and actinide contraction
,” J. Phys. Chem. A
101
, 7128
–7133
(1997
).63.
J. K.
Laerdahl
, K.
Fægri
, Jr., L.
Visscher
, and T.
Saue
, “A fully relativistic Dirac–Hartree–Fock and second-order Möller–Plesset study of the lanthanide and actinide contraction
,” J. Chem. Phys.
109
, 10806
(1998
).64.
C.
Clavaguéra
, J.-P.
Dognon
, and P.
Pyykkö
, “Calculated lanthanide contractions for molecular trihalides and fully Hydrated ions: The contributions from relativity and 4f-shell hybridization
,” Chem. Phys. Lett.
429
, 8
–12
(2006
).65.
T. H.
Dunning
, “Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
,” J. Chem. Phys.
90
, 1007
–1023
(1989
).66.
Q.
Lu
and K. A.
Peterson
, “Correlation consistent basis sets for lanthanides: The atoms La-Lu
,” J. Chem. Phys.
145
, 054111
(2016
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
