Origin Invariant Full Optical Rotation Tensor in the Length Dipole Gauge without London Atomic Orbitals Marco Caricato1, a) Department of Chemistry, University of Kansas, Lawrence, KS 66045 (Dated: 7 April 2021) We present an origin-invariant approach to compute the full optical rotation tensor (Buckingham/Dunn tensor) in the length dipole gauge without recourse to London atomic orbitals, called LG(OI). The LG(OI) approach is simpler and less computationally demanding than the more common LG-London and modified velocity gauge (MVG) approaches and it can be used with any approximate wave function or density functional method. We report an implementation at coupled cluster with single and double excitations level (CCSD), for which we present the first simulations of the origin-invariant Buckingham/Dunn tensor in the length gauge. With this method, we attempt to decouple the effects of electron correlation and basis set incompleteness on the choice of gauge for optical rotation calculations on simple test systems. The simulations show a smooth convergence of the LG(OI) and MVG results with the basis set size towards the complete basis set limit. However, these preliminary results indicate that CCSD may not be close to a complete description of the electron correlation effects on this property even for small molecules, and that basis set incompleteness may be a less important cause of discrepancy between choices of gauge than electron correlation incompleteness. I. INTRODUCTION ing atomic orbitals (GIAOs).6,19,28,29 However, GIAOs cannot be utilized with standard CC methods because Optically active compounds are able to rotate the orbital relaxation is neglected to avoid unphysical poles plane of polarization of impinging light, a phenomenon in the linear response (LR) function due to the reference called optical rotation (OR).1 For isotropic systems (e.g., wave function.21,22 On the other hand, the modified VG a gas or solution phase sample), only a spatially aver- (MVG)17,24 recipe requires the explicit evaluation and aged OR can be measured, often expressed as a normal- removal of the unphysical static limit, and so far it has ized quantity known as specific rotation. However, for been the preferred approach for OR calculations at CC oriented systems such as chiral crystals, one can mea- level.15,16,30,31 sure the OR in a specific direction.2,3 Simulations can in We recently proposed a strategy to overcome the principle provide a direct comparison with experimental origin-dependence issue of LG calculations of specific ro- measurements by evaluating the Buckingham/Dunn op- tation without the complication of London orbitals, an tical activity tensor.4 Such comparison can be used to approach we called LG(OI).32 This is based on a transfor- obtain structure-property relationships, assign the abso- mation of the electric dipole-magnetic dipole polarizabil- lute configuration of the compound, and study the effect ity tensor using the singular value decomposition (SVD) of intermolecular interactions on the optical activity. eigenvectors of the mixed-gauge electric dipole-electric Accurate quantum mechanical (QM) methods based dipole polarizability tensor. The LG(OI) approach is on density functional theory (DFT) and coupled cluster simpler than the LG-GIAOs approach and faster than (CC) theory have been developed for the calculation of the MVG approach, but it shares with the latter the ap- chiroptical properties,5–20 using response theory to eval- plicability to any approximate method. Therefore, we uate the appropriate OR tensor.15,19–24 However, given were able to present the first origin-invariant LG simula- the steep computational scaling of electronic QM calcu- tions of specific rotation with standard CC methods. lations, all of these methods only provide an approxi- In this work, we extend the LG(OI) approach to the mate solution to the Schrödinger equation using an in- calculation of the full Buckingham/Dunn tensor and complete basis expansion for the electron density. Thus, present the first origin-invariant LG-CC simulations of the numerical results depend on the choice of gauge for the full OR tensor. Since we now have two fully origin- the electric dipole and quadrupole operators.15,25–27 Two invariant approaches to evaluate the OR of chiral systems typical choices are the length gauge (LG), which is more at CC level, LG(OI) and MVG, we also explore the ef- intuitive but provides an origin-dependent tensor, and fect of basis set and electron correlation incompleteness. the velocity gauge (VG), with which the OR tensor is We decouple these two sources of approximation using a origin-invariant but has an unphysical static limit.17,24 model system where CC with single and double excita- For the LG and variational methods such as Hartree- tions (CCSD) is exact and another system where CCSD Fock (HF) and DFT, the origin-dependence issue is re- is not exact but we can use fairly large basis sets and solved using London orbitals, also known as gauge includ- extrapolate to the complete basis set limit. The paper is organized as follows: the theory derivation is presented in section II, details of the calculations are reported in sec- tion III, the results of numerical simulations are discussed a)Electronic mail: [email protected]. in section IV, and concluding remarks are summarized in 2 section V. where β and ! are given in atomic units, h̵ is the reduced Planck’s constant (J s), NA is Avogadro’s number, c is the speed of light (m/s), me is the electron rest mass II. THEORY (kg), and M is the molecular mass (amu). In Eq. 6, we used the fact that the A contribution to B is traceless so In this section, we present the theory for the evalua- that Tr( ) = Tr(B) = Tr(β). Therefore, the A tensor is tion of the full OR tensor with the LG(OI) approach.32 In unnecessary in OR simulations of molecules in isotropic order to do that, it is useful to briefly review the equa- media. B tions for the Buckingham/Dunn optical activity tensor The tensors β and A in Eqs. 3-4 are both origin de- , which is defined as:4,23,33 pendent according to: 1 1 R;P B Bαβ = [Tr(B)δαβ − Bαβ] (1) β (O′) = β (O) + α( )d (7) 2 αβ αβ 2 βγδ αγ δ ⎡ ⎤ 1 ⎢ 1 ⎥ ′ 3 (R;R) B = ⎢β + β + Q ( A + A )⎥ (2) Aα,βγ (O ) =Aα,βγ (O) − α dβ αβ ⎢ αβ βα αγδ γ,δβ βγδ γ,δα ⎥ 2 αγ 2 ⎢ 3 δ,γ ⎥ (8) ⎣ ⎦ 3 − α(R;R)d + α(R;R)d δ where is the Levi-Civita operator, β is the electric 2 αβ γ αδ δ βγ dipole-magnetic dipole polarizability tensor, and A is the electric dipole-electric quadrupole polarizability tensor: where a sum over common indices is implied in the last term on the right-hand side of both equations, O is a particular choice of origin of the coordinate system and ⟨ Sµ S ⟩⟨ Sm S ⟩ β = 2 Im 0 α j j β 0 (3) αβ Q 2 2 ′ j≠0 !j − ! O = O + d (9) is a displaced origin, δ is the Kronecker delta tensor, and ⟨ Sµ S ⟩⟨ SΘ S ⟩ A = 2 ! Re 0 α j j βγ 0 (4) α is the electric dipole-electric dipole polarizability tensor α,βγ Q j 2 2 j≠0 !j − ! expressed with two gauge representations for the dipole operator: the superscript (R; R) in Eq. 8 indicates the We use atomic units throughout the paper, except when length gauge representation for both occurrences of the otherwise specified. Since there has been some confu- dipole operator (shown in Eq. 5), while the superscript sion in the literature for the use of the G′ symbol to (R; P ) in Eq. 7 indicates a mixed representation with the indicate the electric dipole-magnetic dipole polarizabil- length gauge for one occurrence of the dipole operator ity with or without the !−1 factor, we use the notation and the velocity gauge for the other one, β = −!−1G′.23 The multipole operators in Eqs. 3-4 are expressed in the length dipole gauge as: µP = −p: (10) µ = −r where p is the momentum operator including an implicit i summation over the number of electrons in the molecule. m = (r × ∇) (5) 2 In an exact calculation, the individual origin dependent 1 2 terms for the two tensors β and A perfectly cancel out Θβγ = − (3rβrγ − δβγ r ) 2 when they are combined in B in Eq. 2 such that the latter which are respectively the electric dipole, the magnetic is origin invariant. However, this is no longer the case in dipole, and the traceless electric quadrupole operators, approximate calculations and the individual elements of with the position r and gradient ∇ operators implicitly B as well as its trace are origin dependent. summed over all the electrons of the molecule. The Greek There are typically two ways to overcome the origin- indices denote Cartesian coordinates, ! is the frequency dependence issue in approximate calculations. For the of the incident electromagnetic radiation while S j⟩ and length gauge and variational methods (like HF and th DFT), one can use London orbitals6,28 (also known as !j are the j excited state wave function and excita- tion frequency, respectively. These definitions are valid GIAOs), but such strategy is not feasible with conven- tional CC methods because the response of the molecu- for non-resonant optical activity (!j ≈~ !) calculations; resonant optical activity is discussed in greater detail lar orbitals is neglected to avoid unphysical poles in the 21 elsewhere.1,4,34,35 response function. An approach that works with any For isotropic media, the observed optical rotation is approximate method is to express the electric multipoles commonly reported as a normalized quantity in units of in the velocity gauge, see Eq.