Comparison of Length, Velocity, and Symmetric Gauges for the Calculation of Absorption and Electric Circular Dichroism Spectra with Real-Time Time-Dependent Density Functional Theory

Abstract

A velocity and symmetric gauge implementation for real-time time-dependent density functional theory (RT-TDDFT) in the package using a Gaussian and plane wave approach is presented, including the explicit gauge-transformed contributions due to the nonlocal part of pseudopotentials. Absorption spectra of gas-phase α-pinene are calculated in length and velocity gauges in the long-wavelength approximation for the application of a δ pulse in linear and full order. The velocity gauge implementation is also applied to a solvated uracil molecule to showcase its use within periodic boundary conditions (PBC). For the calculation of the expectation value of the electric dipole moment in PBC, both the velocity representation and the modern theory of polarization give equivalent absorption spectra if a distributed reference point is used for the nonlocal term of the velocity operator. The discussion of linear response theory takes place in a unified framework in terms of linear response functions in propagator notation, distinguishing the parts of the linear response functions associated with perturbation and response. To further investigate gauge dependence, electric circular dichroism (ECD) spectra of α-pinene were calculated either as magnetic response to an electric field perturbation, in length or velocity gauge, or as electric response to a magnetic field perturbation in the symmetric gauge. Both approaches, electric and magnetic perturbations, have been found to yield equivalent ECD spectra.

The position operator problem in periodic calculations with an emphasis on theoretical spectroscopy

“…For further details on the derivation, as well as the extension to arbitrary unit cell shapes, we refer to the literature. 16,41 The expressions derived by the modern theory of polarization are routinely applied to calculate the expectation value of the bulk polarization, for example in calculations of Raman spectra, 17,32,[42][43][44] Born effective charges and IR spectra, 13,[45][46][47][48][49] ferroelectric materials, 46,[50][51][52][53] piezoelectricity, [54][55][56] dielectric constants, 47,57 the (anomalous) Hall effect, 58,59 magnetoelectric responses, 60 and sum frequency generation spectra. 34,35 As pointed out e.g.…”

“…Another option for the calculation of the expectation value of the electric dipole moment is the velocity gauge. 42 The velocity (or electric current operator) can be calculated as the Heisenberg time derivative of the position operator as [94][95][96]…”

“…For this reason, a complex electron density matrix must be employed to obtain non-vanishing expectation values. This can be achieved, for example, by TD-DFT 42,97 or imaginary perturbations. 98 Additionally, the equality…”

“…94 Special care has to be taken in the evaluation of the commutator of the position operator and the non-local (pseudo)potential operator in eqn ( 16) to ensure correct results. 42,96,99 Similar to the MLWF approach, the application of the velocity dipole operator is not limited to periodic systems but can also be used to remove gauge dependencies in nonperiodic simulations. The velocity gauge operator has been succesfully applied in the calculation of absorption spectra from the X-ray to IR region, 25,42,[100][101][102] photoinduced processes, 103 excited state dipole moments, 100 high harmonic generation, 104,105 electron scattering experiments, 106 optical rotation tensors, 107,108 VCD spectra, 25,26 Electronic Circular Dichroism (ECD) spectra, 42,96 ROA spectra, 33,[109][110][111][112][113] and dielectric constants.…”

“…A perturbation can be applied to the system to recover the effects of the current density j(r). To achieve this, an imaginary correction to the KS-MOs has been applied using Density Functional Perturbation Theory (DFPT), 9,23 and the CA Hamiltonian H CA (see eqn (43)) in combination with velocity-gauge including atomic orbitals (see eqn (42)). 25,98 The basis functions in eqn ( 42) can be Gaussian-type orbitals, Slater-type orbitals, or numerical basis functions.…”