1 Calculation of m olecular g-tensors by sampling spin orientations of generali sed Hartree -Fock states

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Calculation of molecular g-tensors by sampling spin orientations

of generalised Hartree-Fock states

Shadan Ghassemi Tabrizi,1,a,* R. Rodríguez-Guzmán,2 and Carlos A. Jiménez-Hoyos1,†

1Department of Chemistry, Wesleyan University, Middletown, CT 06459, USA

2Department of Applied Physics I, University of Sevilla, Sevilla, E-41011, Spain

aPresent address: Department of Chemistry, University of Potsdam,

Karl-Liebknecht-Str. 24-25, D-14476, Potsdam-Golm, Germany

*shadan_ghassemi@yahoo.com

†cjimenezhoyo@wesleyan.edu

Abstract. The variational inclusion of spin-orbit coupling in self-consistent field (SCF)

calculations requires a generalised two-component framework, which permits the single-

determinant wave function to completely break spin symmetry. The individual components of

the molecular g-tensor are commonly obtained from separate SCF solutions that align the

magnetic moment along one of the three principal tensor axes. However, this strategy raises the

question if energy differences between solutions are relevant, or how convergence is achieved

if the principal axis system is not determined by molecular symmetry. The present work

resolves these issues by a simple two-step procedure akin to the generator coordinate method

(GCM). First, a few generalised Hartree Fock (GHF) solutions are converged, applying, where

needed, a constraint to the orientation of the magnetic-moment or spin vector. Then,

superpositions of GHF determinants are formed through non-orthogonal configuration

interaction. This procedure yields a Kramers doublet for the calculation of the complete g-

tensor. Alternatively, for systems with weak spin-orbit effects, diagonalisation in a basis

spanned by spin rotations of a single GHF determinant affords qualitatively correct g-tensors

by eliminating errors related to spin contamination. For small first-row molecules, these

approaches are evaluated against experimental data and full configuration interaction results. It

is further demonstrated for two systems (a fictitious tetrahedral CH4+ species, and a CuF42–

complex) that a GCM strategy, in contrast to alternative mean-field methods, can correctly

describe the spin-orbit splitting of orbitally-degenerate ground states, which causes large g-

shifts and may lead to negative g-values.

1. Introduction

The g-tensor g is a fundamental quantity in the phenomenological description of electron

paramagnetic resonance (EPR) spectra of molecules with unpaired electrons. The spin

Hamiltonian

ˆS

ˆSB



=  B g S

, (1)

defines the magnetic-field (B) dependent Zeeman splitting of a Kramers doublet, where the

latter is represented by an

pseudospin

[1]. In molecules with light atoms (through the

3d series) the pseudospin is usually perturbatively related to a true electronic spin

in the

absence of spin-orbit coupling (SOC). However, the use of Eq. (1) is not restricted to such

cases [1,2].

To help interpret spectra in terms of electronic structure, quantum-chemical calculations of

g-tensors must consider SOC as the main cause of deviations from the isotropic free-electron

g-value,

2.0023193

g

. From a multitude of computational methods (see, e.g., Refs. [3–5]

for additional literature), we mention the coupled-perturbed treatment of SOC by Hartree Fock

(HF) theory or – more commonly – Kohn-Sham (KS) density functional theory (DFT) [6,7];

generally more accurate ab initio methods include coupled-cluster theory [8], complete active

space self-consistent field (CASSCF) [9,10], and multi-reference configuration interaction

[11,12]. In the spin-orbit state-interaction (SOSI) procedure, a few selected nonrelativistic (or

scalar relativistic) CASSCF solutions form a space for the subsequent diagonalisation of a full

Hamiltonian that includes SOC. The g-tensor is calculated from the resulting Kramers doublet

(see Eq. (4) in the Theory section below) [9,10]. In case of strong mixing between many

nonrelativistic states in systems featuring heavy atoms, SOC should be treated in a one-step

procedure to calculate the Kramers doublet [13]. Dynamic correlation effects are often of

relatively minor importance for g-tensors [9], but may be captured by multi-reference

perturbation theory [5,9,14]. DMRG-CASSCF can handle larger active spaces, thereby

enabling g-tensor calculations in rather large strongly correlated systems [4,15], e.g.,

multinuclear transition-metal clusters [4].

Despite such progress, simple and reliable low-scaling approaches, preferably with a mean-

field cost and applicable to a wide range of systems, are still of high interest. Specifically two-

or four-component (2c, 4c) HF or DFT approaches implicitly take into account higher-order

spin-orbit effects, important for g-tensors in 4d, 5d or actinide complexes, by treating SOC self-

consistently [3,16–19]. A common strategy is to seek SCF solutions for different orientations

of the magnetic moment and to derive each g-tensor component from the respective expectation

value

[20]. Three solutions with

aligned along the principal axes x, y and z, related to

elements of the molecular point-group, afford the principal g-components in rhombic

molecules [20]. This approach is hence called 3SCF [3]. However, for lower molecular

symmetry, up to six different SCF solutions are needed to determine all components,

possibly

with

pointing along the Cartesian axes or bisectors [20], but SCF-convergence onto a

desired

orientation from some initial guess may not be guaranteed.

In addition, the somewhat unsatisfactory fact that each g-component is determined from a

separate SCF solution has raised conceptual questions [21]. In the presence of SOC, the

different solutions are in general not symmetry related and therefore not degenerate. The energy

anisotropy with respect to different orientations of

is thus expected (and was

quantitatively explored for a few

systems in Ref. [22]) and does not demand a vaguely

hypothesised [21] revision of the foundations of EPR spectroscopy. It is however not clear if or

how the anisotropy affects g-tensor predictions from mean-field calculations. Note that a similar

conceptual issue does not arise in other approaches [9,10,23]. Specifically, the calculation of

the physically relevant quantities [2]

=G gg

(Eq. (4) below) and

det( )g

is unequivocal when

based on a qualitatively correct CASSCF-type wave function or, more generally, a basis-set

exact full configuration interaction (FCI) wave function. In other words, the calculation of the

field-dependent splitting, determined by G (in the limit

0→B

), and the prediction of the sense

of precession of the magnetic moment around the field direction, determined by the sign of

det( )g

, are unambiguous.

This work proposes a mean-field approach that is closely related to 3SCF, but conceptually

more satisfactory and of broader applicability. It is based on a manifold of constrained HF

solutions with different orientations of

(or

, or

). We discuss straightforward

-constrained optimisation based on a Thouless parametrisation of GHF-type Slater determinants

along with an optimisation library that can handle nonlinear constraints. Diagonalisation in this

manifold yields a single Kramers doublet that determines all components of a qualitatively

correct g-tensor. Alternatively, a suitable manifold can be spanned by spin rotations of a single

In low-symmetry systems, the g-tensor may in principle have nine independent components. However, it is

always possible to eliminate the three antisymmetric components by pseudospin rotations, see Ref. [1].

GHF solution. Where required, orbital degeneracies can be treated by considering orientational

manifolds associated with several symmetry-broken configurations.

2. Theory and Computations

The spin Hamiltonian of Eq. (1) generically describes a tensorial linear field-induced level

splitting (the lifting of Kramers degeneracy). A connection to electronic-structure theory is

established by identifying the electronic Zeeman term, Eq. (2), as the cause of the splitting,

Zee

ˆˆ

H= − Bμ

. (2)

Let

( , , )

x y z

=μ μ μ μ

denote the

22

matrices of the three Cartesian components of the

electronic magnetic moment

ˆˆ

ˆ()



=+μ S L

in the Kramers-pair basis of time-reversal

conjugate electronic states,



and

   

. Up to a constant factor, the components of g

are the coefficients in an expansion of

( , , )

x y z

=μ μ μ μ

in terms of Pauli spin matrices [2]. The

specific component values depend on the definition of pseudospin functions, meaning the

establishment of a one-to-one correspondence between pseudospin states

 

,

and

electronic states

 

,

[1,2], but the symmetric Abragam-Bleaney tensor [2,24]

=G gg

is an invariant [2] that defines the splitting

E

, Eq. (3),

B m n mn

E B B G



= 

, (3)

resulting from the diagonalisation of

Zee

in the basis

 

,

. The elements

( , , , )m n x y z=

were derived by Gerloch and McMeeking [24],

ˆˆ

mn m n

G v w w v



=  = 

=

. (4)

Diagonalisation of G amounts to a rotation to its principal-axis system. The square roots of the

eigenvalues yield the principal values of the diagonal tensor

diag( , , )

x y z

g g g=g

gG=

although, to be precise, each individual value is only defined up to a sign,

gG=

(see

below). We report shifts

n n e

g g g = −

in ppm

(10 )

−

. The outlined approach, advocated by

Bolvin [9], who also cited earlier applications of Eq. (4), is a standard recipe in conjunction

with SOSI between nonrelativistic (or scalar relativistic) CASSCF solutions [10].

The quantity det(g) is another invariant (unchanged by arbitrary unitary transformations

among pseudospin functions). Its sign corresponds to the sign of the product

x y z

g g g

[1,2],

which defines the sense of precession of

and is relevant for the absorption of circularly

polarised radiation [2]. Although the signs of the individual principal components are arguably

not meaningful for the prediction of any property, they may still be fixed by the following

simple procedure: increase the SOC strength stepwise from zero, where pseudospin is

equivalent to true spin (thus,

g=g

), until a realistic SOC strength is reached, while

successively updating the pseudospin functions by a Bloch [25]/des Cloizeaux [26] perturbative

connection to the respective functions at the previous step. This in essence describes the

adiabatic-connection strategy of Chibotaru [1]; see also Ref. [27] for simple applications in a

different context, including

S

systems. In other cases, high molecular symmetry anchors

the adiabatic connection and fixes the signs of principal g-values [1,28]. However, we presently

do not consider the formal problem of pseudospin definition. In our test set of light molecules,

SOC effects are weak, keeping pseudospin closely related to true spin. Hence, all components

of g are positive. Besides, rhombic symmetry determines the principal axis system.

On the other hand, two artificial tetrahedral model systems (CH4+ and CuF42–) display

orbitally degenerate ground states and serve to demonstrate that a GCM approach (explained

below) can also describe the first-order SOC-induced level-splitting of orbitally degenerate

states, where pseudospin is not anymore related to true spin and negative g-values may ensue.

In these cases, tetrahedral symmetry enforces isotropy,

x y z

g g g==

[28]. The sign of

x y z

g g g

corresponds to the sign of the quotient

x z y

g g g

()

x z x z z x ab

y B y ab

g g i



−

=− μ μ μ μ

, (5)

where a,b = 1,2 is an arbitrary index combination for a non-zero element

()

y ab

of the

22

matrix

. Eq. (5), derived in Ref. [23], adheres to the usual Condon-Shortley phase

convention.

The 3SCF procedure outlined in the Introduction was first applied in the frame of GHF [20]

(including one- and two-electron spin-orbit terms) and later also in quasi-relativistic 2c-

DFT [19,29,30] or 4c-Dirac-KS calculations [17]. Our present focus is on GHF, which breaks

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1Calculationofmolecularg-tensorsbysamplingspinorientationsofgeneralisedHartree-FockstatesShadanGhassemiTabrizi,1,a,*R.Rodríguez-Guzmán,2andCarlosA.Jiménez-Hoyos1,†1DepartmentofChemistry,WesleyanUniversity,Middletown,CT06459,USA2DepartmentofAppliedPhysicsI,UniversityofSevilla,Sevilla,E-41011,SpainaPr...

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1 Calculation of m olecular g-tensors by sampling spin orientations of generali sed Hartree -Fock states

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