Zeroth order regular approximation approach to electric dipole moment interactions of the electron
aa r X i v : . [ phy s i c s . c h e m - ph ] M a r Zeroth order regular approximation approach to electric dipole momentinteractions of the electron
Konstantin Gaul and Robert Berger
Fachbereich Chemie, Philipps-Universit¨at Marburg, Hans-Meerwein-Straße 4, 35032 Marburg,Germany (Dated: March 21, 2017)
A quasi-relativistic two-component approach for an efficient calculation of P , T -odd interactions caused by apermanent electric dipole moment of the electron (eEDM) is presented. The approach uses a (two-component)complex generalized Hartree-Fock (cGHF) and a complex generalized Kohn-Sham (cGKS) scheme withinthe zeroth order regular approximation (ZORA). In applications to select heavy-elemental polar diatomicmolecular radicals, which are promising candidates for an eEDM experiment, the method is compared torelativistic four-component electron-correlation calculations and confirms values for the effective electrical fieldacting on the unpaired electron for RaF, BaF, YbF and HgF. The calculations show that purely relativisticeffects, involving only the lower component of the Dirac bi-spinor, are well described by treating only theupper component explicitly. I. INTRODUCTION
Violations of fundamental symmetries, such as thoserelated to a combined charge conjugation ( C ) and parity( P ) operation ( CP -violation), provide stringent tests forphysics beyond the Standard Model of particle physics,which is often referred to as new physics. The perma-nent electric dipole moment (EDM) of particles, which isthe target of the present work, emerges from violation ofboth parity ( P ) and time-reversal ( T ) symmetry that isrelated to CP -violation via the CPT -theorem . Even before the first experimental evidence of CP -violation in Kaon decays, Salpeter studied the effectof a permanent EDM of an electron (eEDM) and pre-sented first calculations for hydrogen-like atoms in 1958. Sanders found strong relativistic enhancement of effectsdue to an eEDM in the late 1960s and suggestedthe use of polar diatomic molecules that contain a high Z element (with Z being the nuclear charge) for thesearch of the proton EDM. In the 1970s Labzowsky, Gorshkov et. al and Sushkov and Flambaum showed that effects become large in polar heavy diatomicradicals. Some years later Sushkov, Flambaum andKhriplovich suggested to exploit small Ω-doublingeffects in Σ / -ground states of molecules such as HgFor BaF, which were first studied theoretically by Ko-zlov in a semi-empirical model in 1985. Recently, in2014, the currently lowest upper limit on the eEDM( d e . . × − e · cm) was measured using the H ∆ -state of ThO. As CP -violation in the Standard Modelis only embedded on the level of quarks and gives rise toan EDM in the lepton sector only via higher order radia-tive corrections, an eEDM is a sensitive probe for newphysics. The theoretical search for still more favourable can-didates for eEDM experiments, employing relativisticquantum chemistry, is vivid.
So far, besides thestrong Z -dependence of CP -violating effects, there ap-pears to be no thorough understanding of the mecha- nisms that make eEDM enhancement in molecular sys-tems large, but first attempts in this direction havebeen reported recently . Thus, a systematic study of P , T -odd effects in molecules is of avail. Yet relativisticmany-electron calculations are computationally demand-ing. The main effort in relativistic calculations stemsfrom an explicit consideration of the small component ofthe Dirac wave function. Quasi-relativistic approaches,such as the zeroth order regular approximation (ZORA),improve on computation time considerably and performvery well in most molecular calculations. Even in thedescription of relativistic molecular properties such asparity violating effects in molecules ZORA proved to bevery reliable.
II. METHODOLOGYA. Theory of eEDM Interactions in Molecules
Salpeter introduced a perturbation of the Dirac equa-tion due to interactions with an electric dipole momentof an electron to describe permanent EDMs of atoms. The Lorentz- and gauge-invariant formulation appears as h γ µ (cid:16) ı ~ ∂ µ + ec A µ (cid:17) − m e c × i ψ = − d e γ σ µν F µν ψ. (1)Here c is the speed of light in vacuum, e is the electricconstant, m e is the mass of the electron, ı = √− d e is the eEDM, ψ is a Dirac four-spinor, A µ = (cid:16) Φ , − ~A (cid:17) is the four-potential with the scalar andvector potentials Φ and ~A , ∂ µ = ∂∂ x µ is a first derivativeof the four vector x µ = ( ct, x, y, z ), γ µ , γ are the Diracmatrices in standard notation: γ = (cid:18) × × × − × (cid:19) , γ k = (cid:18) × σ k − σ k × (cid:19) γ = ı γ γ γ γ (2)and σ µν = [ γ µ , γ ν ] − /
2, with [ a, b ] − = ab − ba be-ing the commutator. F µν = ∂ µ A ν − ∂ ν A µ is the fieldstrength tensor. In the above equations index notation( µ, ν = 0 , , , k = 1 , ,
3) and Einstein’s sum conven-tion are employed. After evaluation of the tensor product σ µν F µν the left-hand side of Eq. (1) reduces to a termproportional to the four-component analogue of the Paulispin-matrix vector ~ Σ = × ⊗ ~ σ , representing the inter-action with the electric field ~ E and a term proportional toı ~ α , where α k = γ γ k with k = 1 , ,
3, representing theinteraction with the magnetic field ~B . Thus the Lorentzinvariant eEDM Hamiltonian has the formˆ H eEDM = − d e γ h ~ Σ · ~ E + ı ~ α · ~ B i . (3)It was shown that the magnetic term gives minor contri-butions in many-body calculations and in leading orderthis Hamiltonian further reduces to the interaction withthe electric field: ˆ H eEDM ≈ − d e γ ~ Σ · ~ E . (4)In 1963 Schiff stated, that in the non-relativistic limitthe expectation value of ˆ H eEDM of an atom vanishes, in-dependent whether the elementary particles in the atomhave an EDM or not. Therefore an atom in the non-relativistic limit has always a zero EDM. Addition of0 = d e (cid:16) ~ Σ · ~ E ( ~r ) − ~ Σ · ~ E ( ~r ) (cid:17) results in the alternativeformulation ˆ H eEDM = − d e ~ Σ · ~ E ( ~r ) − d e (cid:0) γ − (cid:1) ~ Σ · ~ E ( ~r ) , (5)with the total electrical field ~ E = ~ E int + ~ E ext being a sumof the internal and external fields ~ E int , ~ E ext . Whereas thefirst term on the right is zero due to Schiff’s theorem, thesecond appears only in the small component of the Diracequation, because (cid:0) γ − (cid:1) = − · (cid:18) × × × × (cid:19) . (6)Since the momentum operator commutes with the Dirac-Hamiltonian, the effective Hamiltonian for the eEDM canbe reformulated by commuting the unperturbed Hamil-tonian with a modified momentum operator and definingan effective operator ˆ H d :ˆ H tot = ˆ H + ˆ H eEDM (7)ˆ H eEDM = h ˆ P , ˆ H i + ˆ H d . (8)In consistency with the notation of Lindroth, Lynn andSandars the modified momentum operator of the above derivation is called ˆ P I and the reformulation of ˆ H eEDM using ˆ P I is denoted as Stratagem I, which can be sum-marized as ˆ P I ≡ − ı d e ~ e N elec P i =1 ~ Σ · ˆ ~p i (9) ⇒ ˆ H d , I = − d e N elec P i =1 (cid:0) γ − (cid:1) ~ Σ · ~ E ( ~r i ) , (10)Here ˆ ~p is the linear momentum operator, ~ = h/ (2 π )is the reduced Planck constant and the sum runs overall N elec electrons of system. Additionally, introducing afactor of γ in the modified momentum, an alternativeexpression, called Stratagem II, can be derived: ˆ P II ≡ − ı d e ~ e N elec P i =1 γ ~ Σ · ˆ ~p i (11) ⇒ ˆ H d , II = cd e ~ e N elec P i =1 γ γ ˆ ~p i . (12)This operator has, within the Dirac-Coulomb picture, theadvantage of being a single-particle operator, whereasˆ H d , I is due to the internal electrical field, which is a func-tion of the Coulomb potential V C , a many-body operator(see below). Lindroth, Lynn and Sandars have pointedout, that effects of the Breit interaction and one-photonexchange, which are on the same order, namely O ( α ),give only minor contributions of less than one percent. For the evaluation of ˆ H d , I the mathematical form of theelectrical field is of interest. Starting from its definition ~ E int ( ~r i ) ≡ e ~ ∇ V C ( ~r i ) , (13)using the molecular or atomic Coulomb potential and as-suming a spherically symmetric charge distribution of thenucleus, e.g. a Gaussian charge distribution, the Gaußlaw is valid and the internal electrical field is ~ E ( ~r i ) = N nuc X A =1 Z A e πǫ ~r i − ~r A | ~r i − ~r A | − N elec X j = i e πǫ ~r i − ~r j | ~r i − ~r j | , (14)with the vacuum permittivity ǫ , the number of protons Z A of nucleus A , the vector in position space ~r and thesums running over all N nuc nuclei of the molecule. Thefirst term on the right-hand side arises from the electro-static fields of the nuclei, experienced by the electron,and the second term arises from the electrostatic fieldsof the other electrons. Due to the latter, ˆ H d , I is a many-body Hamiltonian and therefore much more difficult totreat in numerical calculation.Fortunately, it has been shown, that the two-electroncontribution is on the order of one percent only in rel-ativistic many-body calculations and thus it is typicallywell justified to drop this term . The internal electricalfield reduces then to the nuclear contribution only: ~ E ( ~r i ) ≈ N nuc X A =1 Z A e πǫ ~r i − ~r A | ~r i − ~r A | . (15)In this approximation ˆ H d , I is a single-particle operatorand for this reason there is no longer an advantage inusing ˆ H d , II instead, even within the Dirac-Coulomb pic-ture.In this work both forms will be used to derive a quasi-relativistic theory in the zeroth order regular approxi-mation (ZORA) framework. This does not only allow acomparison of the two transformations of the perturbedDirac equation, but also provides a test of the newly de-veloped quasi-relativistic approach, as both stratagemsshould yield approximately the same results. B. Derivation of the eEDM ZORA equations
In the following we will derive the ZORA eEDM inter-action Hamiltonian, starting from the Stratagem I Hamil-tonian ˆ H d , I as perturbation to the molecular Dirac equa-tion (see Equation (10)) and repeat the derivation after-wards for ˆ H d , II , received from Stratagem II (see Equation(12)).
1. Derivation starting from Stratagem I
The ˆ H d , I perturbed molecular Dirac equation can bewritten in block matrix form as (cid:18) ˆ V ( ~r i ) − ǫ c~ σ · ˆ ~π i c~ σ · ˆ ~π i ˆ V ( ~r i ) − ǫ − m e c + 2 d e ~ σ · ~ E ( ~r i ) (cid:19) (cid:18) φ ( ~r i ) χ ( ~r i ) (cid:19) = (cid:18) ~ ~ (cid:19) , (16)where φ , χ are the large and small components of theDirac spinor with energy ǫ shifted by m e c , ˆ ~π = ˆ ~p + e ~A is the minimal coupling relation for the electron with ~A being the vector potential, ˆ V is the scalar potentialenergy operator and i is the index of the electron, whichwill be dropped in the following for better readability.Now using the elimination of small component (ESC)method, the small component can be expressed as χ ( ~r ) = (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) × | {z } A − d e ~ σ · ~ E ( ~r ) | {z } d e B | {z } M − × c (cid:16) ~ σ · ˆ ~π (cid:17) φ ( ~r ) . (17)This equation is valid only if the matrix M is invertible.This is commonly considered to be the case, because B has the form of the Pauli matrices (which are invertible)and A is a diagonal matrix and therefore does not changethe invertibility of B by addition provided that the pa-rameters of the fields involved are assumed to be confinedappropriately.The inversion of the matrix M can now be rewrittenin several ways. The result of the inversion should havethe following properties: (i) it should be divisible into anunperturbed and a perturbed Hamiltonian, (ii) it should be expandable in d e and (iii) the perturbing leading orderterm should be linear in d e .This can be achieved by first extracting A − = (cid:16) m e c − ˆ V ( ~r ) − ǫ (cid:17) − × , using the matrix relation( A + d e B ) − = A − ( × + d e BA − ) − . (18)Now the inverse expression can be expanded in d e whenfor typical field situations (cid:12)(cid:12) d e BA − (cid:12)(cid:12) < × is assumed:( × + d e BA − ) − = ∞ X m =0 (cid:2) − d e BA − (cid:3) m , (19)and we can write down an expression for the ESC Hamil-tonian of infinite order in d e :ˆ H ESCtot,I = ˆ V ( ~r ) × + c (cid:16) ~ σ · ˆ ~π (cid:17) (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − × ∞ X m =0 (cid:20) d e ~ σ · ~ E ( ~r ) (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − (cid:21) m (cid:16) ~ σ · ˆ ~π (cid:17) . (20)This Hamiltonian can be separated into an unperturbedESC-Hamiltonian ˆ H ESC0 ( m = 0 term and potential en-ergy term) and a perturbation due to the eEDM. Thelatter can be reduced to the term linear in d e , as | d e | isvery small.ˆ H ESCtot,I = ˆ H ESC0 + c (cid:16) ~ σ · ˆ ~π (cid:17) (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − d e ~ σ · ~ E ( ~r ) (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − (cid:16) ~ σ · ˆ ~π (cid:17)| {z } ˆ H ESCd , I . (21)For regular approximation we extract (cid:16) m e c − ˆ V ( ~r ) (cid:17) − from (cid:16) m e c − ˆ V ( ~r ) − ǫ (cid:17) − andexpand in the orbital energy ǫ : (cid:16) m e c + ǫ − ˆ V ( ~r ) (cid:17) − = 12 m e c − ˆ V ( ~r ) ∞ X k =0 " − ǫ m e c − ˆ V ( ~r ) k . (22)The ZORA eEDM Hamiltonian, linear in d e , readsˆ H ZORAd , I = c (cid:16) ~ σ · ˆ ~π (cid:17) (cid:16) m e c − ˆ V ( ~r ) (cid:17) − × d e ~ σ · ~ E ( ~r ) (cid:16) m e c − ˆ V ( ~r ) (cid:17) − (cid:16) ~ σ · ˆ ~π (cid:17) . (23)As (cid:20)(cid:16) m e c − ˆ V ( ~r ) (cid:17) − , ~ σ · ~ E ( ~r ) (cid:21) − = 0, a modifiedZORA factor can be defined as ω d , I ( ~r ) = 2 d e c (cid:16) m e c − ˆ V ( ~r ) (cid:17) (24)and the final expression for the ZORA eEDM interactionHamiltonian readsˆ H ZORAd , I = (cid:16) ~ σ · ˆ ~π (cid:17) ω d , I ( ~r ) ~ σ · ~ E ( ~r ) (cid:16) ~ σ · ˆ ~π (cid:17) . (25)This operator is approximately a one-electron operator(see Eq. (15)). From here on we drop now terms de-pending on the vector potential (thus assuming no mag-netic interactions). Then the matrix elements within aone-electron basis set { ϕ λ } are of the form H ZORAd , I ,λρ = D ϕ λ (cid:12)(cid:12)(cid:12) ˆ H ZORAd , I (cid:12)(cid:12)(cid:12) ϕ ρ E = D ϕ λ (cid:12)(cid:12)(cid:12) (cid:16) ~ σ · ˆ ~p (cid:17) ω d , I ( ~r ) ~ σ · ~ E ( ~r ) (cid:16) ~ σ · ˆ ~p (cid:17) (cid:12)(cid:12)(cid:12) ϕ ρ E . (26)The momentum operators in position space are differ-ential operators and therefore make simplifications com-plicated. These operators are hermitian and using thisproperty, we let the left operator act on the left basisfunction and the right on the right basis function to re-ceive the integral H ZORAd , I ,λρ = Z d ~r ( ~ σ · ~p λ ) ∗ ω d , I ( ~r ) ~ σ · ~ E ( ~r ) ( ~ σ · ~p ρ ) , (27)where the momentum operator acting on the basis func-tion was simplified as ˆ ~pϕ λ = ~p λ , which is no longer an op-erator. Hence the commutators of all appearing elementsin the above product are zero: h ( ~ σ · ~p ρ ) , (cid:16) ~ σ · ~ E (cid:17)i − = h ω d , I ( ~r ) , (cid:16) ~ σ · ~ E (cid:17)i − = [ ω d , I ( ~r ) , ( ~ σ · ~p ρ )] − = 0. Using theDirac relation( ~ σ · ~v ) ( ~ σ · ~u ) = ~v · ~u × + ı ~ σ · ( ~v × ~u ) (28) twice (for spin-independent ~u and ~v ), the interaction ma-trix element is divided in three parts, that is H ZORAd , I ,λρ = Z d ~r h ı ( ~p λ ) ∗ ω d , I ( ~r ) · (cid:16) ~ E ( ~r ) × ~p ρ (cid:17) × + ( ~ σ · ~p λ ) ∗ ω d , I ( ~r ) ~ E ( ~r ) · ~p ρ − ~ σ · (cid:16) ( ~p λ ) ∗ ω d , I ( ~r ) × (cid:16) ~ E ( ~r ) × ~p ρ (cid:17)(cid:17)i . (29)This expression can be further simplified using the Graß-mann identity ~a × ( ~b × ~c ) = ~b ( ~a · ~c ) − ~c ( ~a · ~b ), receiving H ZORAd , I ,λρ = ~ Z d ~r h ı ~ ∇ ∗ λ ω d , I ( ~r ) · (cid:16) ~ E ( ~r ) × ~ ∇ ρ (cid:17) + (cid:16) ~ σ · ~ ∇ ∗ λ (cid:17) ω d , I ( ~r ) (cid:16) ~ E ( ~r ) · ~ ∇ ρ (cid:17) − ~ ∇ ∗ λ ω d , I ( ~r ) (cid:16) ~ σ · ~ E ( ~r ) (cid:17) ~ ∇ ρ + (cid:16) ~ ∇ ∗ λ · ~ E ( ~r ) (cid:17) ω d , I ( ~r ) (cid:16) ~ σ · ~ ∇ ρ (cid:17)i , (30)where the momentum operator ˆ ~p = − ı ~ ~ ∇ was insertedand the notation ~ ∇ λ = ~ ∇ ϕ λ ; ~ ∇ ∗ λ = ~ ∇ ϕ ∗ λ (31)was introduced. Now we can separate terms with respectto the spatial components of the spin and write the eEDMHamiltonian in the formˆ O = ˆ O (0) + σ ˆ O (1) + σ ˆ O (2) + σ ˆ O (3) . (32)This results in spin-free matrix-elements of the ZORAeEDM Hamiltonian H ZORA , (0)d , I ,λρ = ı ~ Z d ~r ~ ∇ ∗ λ ω d , I ( ~r ) · (cid:16) ~ E ( ~r ) × ~ ∇ ρ (cid:17) (33)and matrix-elements corresponding to the three spatialdirections of spin k = m = l ∧ k, m, l ∈ { , , } H ZORA , ( k )d , I ,λρ = ~ Z d ~r h ∂ ∗ k λ ω d , I ( ~r ) (cid:16) ~ E ( ~r ) · ~ ∇ ρ (cid:17) − ~ ∇ ∗ λ ω d , I ( ~r ) E k ( ~r ) ~ ∇ ρ + (cid:16) ~ ∇ ∗ λ · ~ E ( ~r ) (cid:17) ω d , I ( ~r ) ∂ k ρ i , (34)where the notation ∂ k λ = ∂ k ϕ λ ; ∂ ∗ k λ = ∂ k ϕ ∗ λ (35)was introduced. The internal electrical field is calculatedin the approximation of Eq. (15) and in the modifiedeEDM ZORA factor a model potential, introduced by vanW¨ullen, is used to alleviate the gauge dependence ofZORA. With that the matrix elements are implementedas H ZORA , (0)d , I ,λρ = 2ı ~ ed e c πǫ N nuc X α =1 Z d ~r · Z α (cid:16) m e c − ˜ V ( ~r ) (cid:17) (( x − x α ) + ( y − y α ) + ( z − z α ) ) / × (cid:0) ( x − x α ) (cid:0) ∂ ∗ z λ ∂ y ρ − ∂ ∗ y λ ∂ z ρ (cid:1) + ( y − y α ) (cid:0) ∂ ∗ x λ ∂ z ρ − ∂ ∗ z λ ∂ x ρ (cid:1) + ( z − z α )) (cid:0) ∂ ∗ y λ ∂ x ρ − ∂ ∗ x λ ∂ y ρ (cid:1)(cid:1)(cid:3) (36) H ZORA , ( k )d , I ,λρ = 2 ~ ed e c πǫ N nuc X α =1 Z d ~r · Z α (cid:16) m e c − ˜ V ( ~r ) (cid:17) (( x − x α ) + ( y − y α ) + ( z − z α ) ) / × (cid:0) (x k − x k,α ) (cid:0) ∂ ∗ k λ ∂ k ρ − ∂ ∗ l λ ∂ l ρ − ∂ ∗ m λ ∂ m ρ (cid:1) + (x l − x l,α ) · (cid:0) ∂ ∗ l λ ∂ k ρ + ∂ ∗ k λ ∂ l ρ (cid:1) + (x m − x m,α ) · (cid:0) ∂ ∗ m λ ∂ k ρ + ∂ ∗ k λ ∂ m ρ (cid:1)(cid:1)(cid:3) (37)These integrals can not be solved analytically and there-fore numerical integration on a Becke grid is used forcalculations of the eEDM matrix elements. Finally theeEDM interaction can be calculated using the corre-sponding density and spin-density matrices D ( µ ) , whichare obtained from the molecular orbital coefficients C ( α ) λ , C ( β ) λ of orbital λ for up- ( α ) and down-spin ( β )by the formulas D (0) λρ = N occ X i =1 h(cid:16) C ( α ) λi (cid:17) ∗ C ( α ) ρi + (cid:16) C ( β ) λi (cid:17) ∗ C ( β ) ρi i (38a) D (1) λρ = N occ X i =1 h(cid:16) C ( α ) λi (cid:17) ∗ C ( β ) ρi + (cid:16) C ( β ) λi (cid:17) ∗ C ( α ) ρi i (38b) D (2) λρ = − ı N occ X i =1 h(cid:16) C ( α ) λi (cid:17) ∗ C ( β ) ρi − (cid:16) C ( β ) λi (cid:17) ∗ C ( α ) ρi i (38c) D (3) λρ = N occ X i =1 h(cid:16) C ( α ) λi (cid:17) ∗ C ( α ) ρi − (cid:16) C ( β ) λi (cid:17) ∗ C ( β ) ρi i , (38d)where the sum runs over all N occ occupied orbitals i .Then the molecular expectation value of the eEDM in-teraction Hamiltonian reads H ZORAd , I = X λ,ρ h − Im n D (0) λρ o Im n H ZORA , (0)d , I ,λρ o + Re n D (1) λρ o H ZORA , (1)d , I ,λρ + Re n D (2) λρ o H ZORA , (2)d , I ,λρ + Re n D (3) λρ o H ZORA , (3)d , I ,λρ i . (39)These formulas are the working equations used to cal-culate the expectation value of H ZORAd , I in first or- der perturbation theory. In the present calculationsthe (spin-)density matrices are obtained from a self-consistent field (SCF) calculation within the complexgeneralized Hartree-Fock (cGHF) or complex generalizedKohn-Sham (cGKS) formalism using the ZORA Hamil-tonian, but could in principle be also obtained from amore sophisticated electronic structure treatment.
2. Derivation starting from Stratagem II
With the perturbation ˆ H d , II the block matrix form ofthe molecular Dirac equation looks like ˆ V ( ~r i ) − ǫ c (cid:16) ~ σ · ˆ ~π i + d e e ~ ˆ ~p i (cid:17) c (cid:16) ~ σ · ˆ ~π i − d e e ~ ˆ ~p i (cid:17) ˆ V ( ~r i ) − ǫ − m e c (cid:18) φ ( ~r i ) χ ( ~r i ) (cid:19) = (cid:18) ~ ~ (cid:19) . (40)Again in the following the electronic index will bedropped. ESC yields for the small component χ ( ~r ) = (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − c (cid:18) ~ σ · ˆ ~π − d e e ~ ˆ ~p (cid:19) φ ( ~r ) . (41)Now the perturbation only appears in the numerator andno special considerations respective to the denominator(in parentheses) are needed. Hence the ESC Hamiltoniancan be written down asˆ H ESCtot,II = ˆ V ( ~r ) + c (cid:18) ~ σ · ˆ ~π + 2ı d e e ~ ˆ ~p (cid:19) × (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − (cid:18) ~ σ · ˆ ~π − d e e ~ ˆ ~p (cid:19) . (42)Expanding the second term on the right one obtains anunperturbed ESC Hamiltonian, a perturbations linear in d e and a quadratic term in d e :ˆ H ESCtot,II = ˆ H ESC0 + c d e ~ ˆ ~p (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − ˆ ~p | {z } ˆ H ESCII ( d ) + c d e e ~ ˆ ~p (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − (cid:16) ~ σ · ˆ ~π (cid:17) − c (cid:16) ~ σ · ˆ ~π i (cid:17) (cid:16) m e c − ˆ V ( ~r ) + ǫ (cid:17) − d e e ~ ˆ ~p | {z } ˆ H ESCII ( d e ) . (43)When considering the terms in first order of d e only andcarrying out the ZORA expansion, one obtains the re-sulting eEDM interaction Hamiltonian as (again with thevector potential depending terms dropped)ˆ H ZORAd , II = ıˆ ~p ω d , II ( ~r ) (cid:16) ~ σ · ˆ ~p (cid:17) − ı (cid:16) ~ σ · ˆ ~p (cid:17) ω d , II ( ~r )ˆ ~p , (44)where the modified ZORA factor is defined as ω d , II ( ~r ) = 2 d e c e ~ m e c − e ~ ˆ V ( ~r ) . (45)Its matrix elements in an arbitrary one-electron basis { ϕ λ } are H ZORAd , II ,λρ = D ϕ λ (cid:12)(cid:12)(cid:12) ˆ H ZORAd , II (cid:12)(cid:12)(cid:12) ϕ ρ E = D ϕ λ (cid:12)(cid:12)(cid:12) ıˆ ~p ω d , II ( ~r ) (cid:16) ~ σ · ˆ ~p (cid:17) − ı (cid:16) ~ σ · ˆ ~p (cid:17) ω d , II ( ~r )ˆ ~p (cid:12)(cid:12)(cid:12) ϕ ρ E . (46)Due to the simple form of the operator we can directlyproceed in writing the matrix elements of ˆ H ZORAd , II interms of Eq. (32) and get in analogy to (37) H ZORA , ( k )d , II ,λρ = 2 ~ d e c e Z d ~r ∂ ∗ k λ ~ ∇ ρ + (cid:16) ~ ∇ λ (cid:17) ∗ ∂ k ρ m e c − ˜ V ( ~r ) , (47)and there is no spin-free part in the Hamiltonian. Againthe integrals are evaluated numerically due to the appear-ance of the model potential ˜ V ( ~r ) in the denominator. Fi-nally, the total ZORA eEDM interaction energy derivedfrom Stratagem II is evaluated from the (spin-)densitymatrices as presented in Eq. (39) for ˆ H ZORAd , I . III. COMPUTATIONAL DETAILS
For the calculation of two-component wave functions atthe GHF-/GKS-ZORA level a modified version of the quantum chemistry program package Turbomole was used. In order to calculate the P , T -odd eEDM inter-action, the program was extended with the ZORA eEDMHamiltonians implemented as shown in Equations (36),(37), (39) and (47).For density functional theory (DFT) calculationswithin the Kohn-Sham framework the hybrid Becke three parameter exchange functional and Lee, Yang and Parrcorrelation functional (B3LYP) was employed. Forall calculations an atom centered basis set of 37 s, 34 p,14 d and 9 f uncontracted Gaussian functions with the ex-ponential coefficients α i composed as an even-temperedseries as α i = a · b N − i ; i = 1 , . . . , N , with b = 2 fors- and p-function and with b = (5 / / × / ≈ . P -violating interactions in heavy polar di-atomic molecules. The basis set centered at the flu-orine atom was represented by an uncontracted atomicnatural orbital (ANO) basis of triple- ζ quality . TheZORA-model potential ˜ V ( ~r ) was employed with addi-tional damping as proposed by van W¨ullen. For the calculations of two-component wave func-tions and properties a finite nucleus was used, describedby a spherical Gaussian charge distribution ρ α ( ~r ) = ρ e − ζα ~r , where ρ = eZ (cid:16) πζ α (cid:17) / and the root meansquare radius ζ α of nucleus α was used as suggested byVisscher and Dyall. The mass numbers A were chosenas nearest integer to the standard relative atomic mass,i.e. F, Ba,
Yb,
Hg,
Ra.The nuclear distances were optimized at the levels ofGHF-ZORA and GKS-ZORA/B3LYP, respectively. Forstructure optimizations at the DFT level the nucleus wasapproximated as a point charge. The distances obtainedare given in the results section.
IV. RESULTS AND DISCUSSION
In the following we focus on select diatomic moleculeswith a Σ / -ground state, which have been studiedextensively in literature , namely BaF,YbF, HgF and RaF. These systems are well suited forthe validation of the here presented ZORA approach.The eEDM contribution to the effective spin-rotationalHamiltonian for diatomic molecules in a Σ / -state hasthe form H sr = d e W d Ω , (48)where Ω = ~J e · ~λ is the projection of the total angularmomentum of the electron ~J e on the molecular axis, de-fined by the unit vector ~λ pointing from the heavy to thelight nucleus and W d = D ˜ ψ (cid:12)(cid:12)(cid:12) ˆ H d (cid:12)(cid:12)(cid:12) ˜ ψ E d e Ω , (49)with the ZORA wave function ˜ ψ . In some publications,which will be referred to in the following, instead of W d only the effective electrical field E eff = W d Ω, was re-ported. In the tables below, we have converted thesevalues for comparison then to W d using Ω = 1 / a for YbF,3.33 a for HgH, 3.91 a for HgF, 4.11 a for BaF and4.26 a for RaF; and Ω was in the GKS framework: 0.473for YbF, 0.497 for HgF and 0.500 for BaF and RaF. TheGHF inter-nuclear distances are: 3.90 a for YbF, 3.30 a for HgH, 3.82 a for HgF, 4.16 a for BaF and 4.30 a forRaF; and Ω was in the GHF framework: 0.498 for HgFand 0.500 for YbF,BaF and RaF. A. eEDM enhancement in the Σ / -ground states of BaFand RaF We start our discussion with BaF, which was stud-ied already in the beginnings of the search for molecular P , T -violation. A number of calculations of W d or W d Ωto compare to exists for this open-shell diatomic moleculein the literature.From Table I on page 7 we see that the results calcu-lated with the two different ZORA operators are equalwithin the given precision. This suggests that differencesin the transformations made to obtain H d are of onlyminor importance within ZORA.Early calculations with the generalized relativistic ef-fective core potential (GRECP) method without effectiveoperator (EO) based perturbative corrections were signif-icantly lower in magnitude than Kozlov’s semi-empiricalestimates. Changes upon inclusion of EO based correc-tions imply that the inclusion of spin-polarization is cru-cial for a good description of P , T -odd properties.The most recent four-component Dirac–(Hartree)–Fock (DF) based restricted active space (RAS) configu-ration interaction (DF-RASCI) calculations are in a verygood agreement with GRECP calculations at the highestlevel of theory (RASSCF/EO, with RASSCF meaning re-stricted active space self-consistent field) and are also ingood agreement with the semi-empirical results. The GHF- and GKS-ZORA calculations of this workare rather compared to electron-correlation calcula-tions than (paired) DF, since the complex GHF/GKSapproach already includes spin-polarization effects.The GHF results are in a very good agreement( ∼ W d ( ∼ W d ,although not explicitly considering the small component.Recent publications called attention to RaF as apromising candidate for the first measurement of P - and P , T -odd effects in molecules. Table I. Comparison of literature data of the P , T -odd eEDM interaction parameter W d of the spin-rotational Hamiltonian of BaF calculated with differ-ent four-component methods and with a quasi-relativisticGHF/GKS-ZORA approach.Method W d e · cm10 Hz · h Exp.+SE (Ref. 59 and 72) − . (Ref. 59 and 73) − . (Ref. 60) − . (Ref. 60) − . (Ref. 60) − . (Ref. 60) − . (Ref. 67) − . (Ref. 67) − . (Ref. 71) − . W d , I W d , II GHF-ZORA (this work) − . − . − . − . Semi-empirical estimates of W d calculated from experimen-tal hyperfine coupling constants. Generalized relativistic effective core potential two-step ap-proach without electron-correlation calculations (a), witheffective operator technique based many-body perturbationtheory of second order (b), with restricted active space SCFelectron-correlation calculation (c) and with both (d). Dirac-Fock calculation without electron-correlation (a),with electron-correlation effects on the level of restrictedactive space CI (b). Multi-reference CI calculation within a non-relativisticframework (estimated spin-orbit energy).
Table II. Comparison of literature data of the P , T -odd eEDMinteraction parameter W d of the spin-rotational Hamiltonianof RaF calculated with different four-component methods andwith a quasi-relativistic GHF/GKS-ZORA approach.Method W d e · cm10 Hz · h SODCI (Ref. 23) − . basis +∆ triples2 (Ref. 23) − . (Ref. 29) − . W d , I W d , II GHF-ZORA (this work) − . − . − . − . Spin-orbit direct configuration interaction approach Relativistic two-component Fock-space coupled-cluster ap-proach with single and double amplitudes (CCSD) with basisset corrections from CCSD calculations with normal and largesized basis sets and triple-cluster corrections from CCSD calcu-lations with and without perturbative triples. Dirac-Fock calculation with electron-correlation effects on thelevel of coupled cluster with single and double excitations (c).
For RaF both methods, GHF and GKS, give resultsthat are in good agreement with those of three types ofrelativistic electron-correlation calculations (both below10% deviation, see Table II on page 7) with the DFTresults agreeing slightly better with the literature val-ues (below 5% error). Electron correlation effects as de-scribed on the DFT level display the same trends as ob-served for BaF.
B. eEDM enhancement in the Σ / -ground state of YbF YbF is probably the best studied molecule with respectto molecular CP -violation. The values of W d , calculated Table III. Comparison of literature data of the P , T -odd eEDMinteraction parameter W d of the spin-rotational Hamiltonianof YbF calculated with different methods and with a quasi-relativistic GHF/GKS-ZORA approach.Method W d e · cm10 Hz · h Exp.+SE+corr. (Ref. 58, 60, and 74) − . (Ref. 61) − . (Ref. 61) − . (Ref. 62) − . f (Ref. 62) − . (Ref. 63) − . (Ref. 64) − . (Ref. 66) − . (Ref. 66) − . (Ref. 68) − . (Ref. 69) − . (Ref. 70) − . (Ref. 70) − . (Ref. 71) − W d , I W d , II GHF-ZORA (this work) − . − . − . − . Semi-empirical estimates of W d calculated from experimental hy-perfine coupling constants with correction of higher sphericalwaves (b). Generalized relativistic effective core potential two-step approach(a), with restricted active space SCF electron-correlation calcu-lation without (b), with (c) effective operator technique basedmany-body perturbation theory of second order and (d) addi-tional 4 f -hole corrections. Restricted DHF with core-polarization corrections. Since the val-ues are approximately by a factor of two lower, it is likely that adifferent definition of W d was used. Unrestricted Dirac-Fock all-electron calculation. Dirac-Fock calculation without electron-correlation (a), withelectron-correlation effects on the level of restricted active spaceCI (b, improved calculations: d), with electron-correlation ef-fects on the level of second order perturbation theory (c), withelectron-correlation effects on the level of coupled cluster withsingle and double excitations (e). Multi-reference CI calculation within a non-relativistic frame-work (estimated spin-orbit energy). via Eq. (10) and via Eq. (12) are in an excellent agree-ment and deviations are smaller than one percent (seeTable III on page 8). This confirms the approximate equivalence of the two used stratagems to calculate W d within ZORA.As can be seen in Table III on page 8 there is a largediscrepancy between the literature results with devia-tions of up to 30%. Whereas calculations with Kozlov’ssemi-empirical model predict rather large values for themagnitude of W d (about 12 . ×
24 Hz · he · cm ), earlyGRECP/RASSCF calculations and Dirac–Fock calcula-tions without consideration of electron-correlation yieldmuch lower absolute values. This may result fromthe neglect of spin-polarization effects, which play a ma-jor role, as has been discussed in the previous section.Yet, more recent four-component electron-correlationcalculations and GRECP calculations with perturba-tive effective operator corrections, which include spin-polarization, show a better agreement and can betaken as the most reliable of the shown literaturevalues.
In comparison to DF, concerning the parameter W d the deviation of the here presented ZORA approach isbelow 5% for GHF and of the order of 10% for GKS.Restricted DF calculations deviate almost by a factorof two from these results and therefore it can be assumedthat a different definition of the effective Hamiltonian wasused. Calculations within a non-relativistic framework,reported in Ref. 71 overestimate the magnitude of W d by about a factor of two and are not reliable (see alsoTable IV on page 9).Again, electron correlation corrections as estimated onthe DFT level lower the absolute value of W d . C. eEDM enhancement in the Σ / -ground state of HgF Although not studied in such detail as YbF, there is agood amount of literature on HgF as well, partially dat-ing back to the 1980s. Furthermore the calculations,which will be discussed in the following, show that mer-cury compounds can provide large enhancements of P -and CP -violation and therefore are very interesting tostudy.As for the other compounds discussed the agreementbetween W d , I and W d , II is excellent, confirming the valid-ity of the two transformations of the eEDM Hamiltonianin the ZORA picture.Whereas GRECP/RASSCF and DF results are in linewith the semi-empirical estimates by Kozlov, althoughspin-polarization effects are not accounted for , morerecent four-component relativistic coupled cluster calcu-lations predict about 10% larger absolute values for W d .As opposed to the trends observed for the moleculesdiscussed before, for HgF the GHF-ZORA approachoverestimates the magnitude of the results from four-component relativistic electron-correlation calculationsby more than 15%. This appears to be caused by avery pronounced energetic splitting in the Kramers pairstructure of the valence orbitals below the singly occu-pied orbital, which have σ -symmetry. This splitting is Table IV. Comparison of literature data of the P , T -oddeEDM interaction parameter W d of the spin-rotationalHamiltonian of HgF calculated with different meth-ods and with a quasi-relativistic GHF/GKS-ZORA ap-proach.Method W d e · cm10 Hz · h Exp.+SE (Ref. 19, 57, and 76) − (Ref. 57) − (Ref. 71) − (Ref. 25) − . (Ref. 25) − . W d , I W d , II GHF-ZORA (this work) − . − . − . − . Semi-empirical estimates of W d calculated from experi-mental hyperfine coupling constants. Generalized relativistic effective core potential two-stepapproach, with restricted active space SCF electron-correlation calculation without (a), with (b) effective op-erator technique based many-body perturbation theory ofsecond order and (c) additional 4 f -hole corrections . Multi-reference CI calculation within a non-relativisticframework (estimated spin-orbit energy). Dirac-Fock calculation without electron-correlation (a),with electron-correlation effects on the level of coupledcluster with single and double excitations (b). much smaller in YbF, RaF and BaF.The GKS-ZORA results instead appear to be muchcloser to the literature data (about 8% deviation fromDF-CCSD and even less from DF or GRECP/RASSCF).Here the additional electron-correlation effects as esti-mated on the DFT level, which lead to a reduction ofthe absolute value of W d in comparison to GHF, playa much more important role than for YbF within theGHF/GKS-ZORA approach and the difference betweenthe GHF and GKS results is much larger for HgF.In four-component calculations the electron correlationeffects seem to be less pronounced, although accountingfor spin-polarization leads to very different results whenone compares results of GRECP/RASSCF with those ofDF-CCSD. This may be caused by partial cancellation ofspin-polarization with other correlation effects. V. CONCLUSION
In this paper we derived a ZORA-based perturbationHamiltonian for the description of P , T -odd interactionsdue to an electron electric dipole moment in molecules.With calculations of promising candidates for a search ofan eEDM, we could show that a quasi-relativistic ZORAapproach is well suited for the calculation of a purelyrelativistic effect, although the small component of thewave function is not considered explicitly. The accuracyfor prediction of W d is estimated to be on the order ofabout 20 % for the Σ / -ground state molecules studiedherein, if one considers recent results obtained with elec-tron correlation approaches as a benchmark. This level of accuracy is presently fully sufficient for the identificationof molecular candidate systems for an eEDM search.With the quasi-relativistic approach presented in thiswork an efficient calculation of the eEDM enhancementin molecules is possible. In future work we will study alarger number of molecules with this approach in order toachieve a deeper understanding of the mechanisms thatlead to sizeable P , T -odd properties in molecules. ACKNOWLEDGMENTS
We thank Timur Isaev for discussions. Financial sup-port by the State Initiative for the Development of Scien-tific and Economic Excellence (LOEWE) in the LOEWE-Focus ELCH and computer time by the center for scien-tific computing (CSC) Frankfurt are gratefully acknowl-edged.
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