Andreas J. Thorvaldsen

The Dalton quantum chemistry program system

Dalton is a powerful general‐purpose program system for the study of molecular electronic structure at the Hartree–Fock, Kohn–Sham, multiconfigurational self‐consistent‐field, Møller–Plesset, configuration‐interaction, and coupled‐cluster levels of theory. Apart from the total energy, a wide variety of molecular properties may be calculated using these electronic‐structure models. Molecular gradients and Hessians are available for geometry optimizations, molecular dynamics, and vibrational studies, whereas magnetic resonance and optical activity can be studied in a gauge‐origin‐invariant manner. Frequency‐dependent molecular properties can be calculated using linear, quadratic, and cubic response theory. A large number of singlet and triplet perturbation operators are available for the study of one‐, two‐, and three‐photon processes. Environmental effects may be included using various dielectric‐medium and quantum‐mechanics/molecular‐mechanics models. Large molecules may be studied using linear‐scaling and massively parallel algorithms. Dalton is distributed at no cost from http://www.daltonprogram.org for a number of UNIX platforms.

A density matrix-based quasienergy formulation of the Kohn–Sham density functional response theory using perturbation- and time-dependent basis sets

A general method is presented for the calculation of molecular properties to arbitrary order at the Kohn-Sham density functional level of theory. The quasienergy and Lagrangian formalisms are combined to derive response functions and their residues by straightforward differentiation of the quasienergy derivative Lagrangian using the elements of the density matrix in the atomic orbital representation as variational parameters. Response functions and response equations are expressed in the atomic orbital basis, allowing recent advances in the field of linear-scaling methodology to be used. Time-dependent and static perturbations are treated on an equal footing, and atomic basis sets that depend on the applied frequency-dependent perturbations may be used, e.g., frequency-dependent London atomic orbitals. The 2n+1 rule may be applied if computationally favorable, but alternative formulations using higher-order perturbed density matrices are also derived. These may be advantageous in order to minimize the number of response equations that needs to be solved, for instance, when one of the perturbations has many components, as is the case for the first-order geometrical derivative of the hyperpolarizability.

Arbitrary-Order Density Functional Response Theory from Automatic Differentiation

We demonstrate how the functional derivatives appearing in perturbative time-dependent density functional theory can be calculated using automatic differentiation. The approach starts from a computer implementation of the exchange-correlation energy functional, from which arbitrary-order derivatives are generated automatically. Automatic differentiation is shown to provide an accurate, general, and efficient implementation of higher-order exchange-correlation functional derivatives that is easy to maintain. When used in combination with an arbitrary-order response solver, the methodology allows us to generate arbitrary-order response functions from time-dependent density functional theory.

Theoretical approaches to the calculation of Raman optical activity spectra

In this article, we will give a brief account of the different approaches that have been presented in the literature for calculating Raman optical activity (ROA) spectra by ab initio methods. We will also outline the general structure of a self-consistent-field-based approach for analytic calculations of ROA spectra, including also contributions from London orbitals. The use of London orbitals ensures that the relevant ROA parameters are gauge origin independent. We will also give an outlook on the future of ab initio calculations of Raman optical activity spectra.

Efficient elimination of response parameters in molecular property calculations for variational and nonvariational energies

A general method is presented for the efficient elimination of response parameters in molecular property calculations for variational and nonvariational energies. For variational energies, Wigners 2n+1 rule is obtained as a special case of the more general k(2n+1) rule, which states that for a subset of k perturbations within a total set of z>or=k perturbations, response parameters may be eliminated according to the 2n+1 rule (normally applied to the full set of perturbations). Nonvariational energies may be treated by introducing Lagrange multipliers that satisfy the stronger 2n+2 rule for the k perturbations, while the wave-function parameters still satisfy the 2n+1 rule for the k perturbations. The corresponding rule for nonvariational energies is referred to as the k(2n+1,2n+2) rule. For k=z, the well-known 2n+2 rule for the multipliers is reproduced, while the wave-function parameters satisfy the 2n+1 rule. The application of the k(2n+1) and k(2n+1,2n+2) rules minimizes the total number of response equations to be solved when the molecular property contains k extensive perturbations (e.g., geometrical derivatives) and z-k intensive perturbations (e.g., electric fields).

Gauge-Origin Independent Formulation and Implementation of Magneto-Optical Activity within Atomic-Orbital-Density Based Hartree-Fock and Kohn-Sham Response Theories.

A Lagrangian approach has been used to derive gauge-origin independent expressions for two properties that rationalize magneto-optical activity, namely the Verdet constant V(ω) of the Faraday effect and the ℬ term of magnetic circular dichroism. The approach is expressed in terms of an atomic-orbital density-matrix based formulation of response theory and use London atomic orbitals to parametrize the magnetic field dependence. It yields a computational procedure which is both gauge-origin independent and suitable for linear-scaling at the level of time-dependent Hartree-Fock and density functional theory. The formulation includes a modified preconditioned conjugated gradient algorithm, which projects out the excited state component from the solution to the linear response equation. This is required when solving one of the response equations for the determination of the ℬ term and divergence is encountered if this component is not projected out. Illustrative results are reported for the Verdet constant of H2, HF, CO, N2O, and CH3CH2CH3 and for the ℬ term of pyrimidine, phosphabenzene, and pyridine. The results are benchmarked against gauge-origin independent CCSD values.

Analytic calculations of vibrational hyperpolarizabilities in the atomic orbital basis.

We present an analytic scheme for the calculation of pure vibrational contributions to linear and nonlinear optical properties such as the polarizability and the first and second hyperpolarizabilities. The formalism is fully expressed in terms of a perturbation- and time-dependent atomic orbital basis, using the elements of the density matrix in the atomic orbital basis as the basic variables. We calculate perturbed densities up to third order with respect to the electric field in accordance with the n + 1 rule, and the approach is therefore applicable for the calculation of pure vibrational contributions involving all vibrational coordinates in large molecular complexes. In the case of static electric fields, we therefore only need to calculate 19 response equations, independent of the size of the molecule. If we can determine the molecular energy and force field, the calculation of pure vibrational contributions to the nonlinear optical properties of the molecule is therefore a rather straightforward task. We illustrate the implementation by calculating pure vibrational contributions to the first and second hyperpolarizabilities of molecules containing up to 66 atoms using basis sets of good quality.

Analytic cubic and quartic force fields using density-functional theory

We present the first analytic implementation of cubic and quartic force constants at the level of Kohn-Sham density-functional theory. The implementation is based on an open-ended formalism for the evaluation of energy derivatives in an atomic-orbital basis. The implementation relies on the availability of open-ended codes for evaluation of one- and two-electron integrals differentiated with respect to nuclear displacements as well as automatic differentiation of the exchange-correlation kernels. We use generalized second-order vibrational perturbation theory to calculate the fundamental frequencies of methane, ethane, benzene, and aniline, comparing B3LYP, BLYP, and Hartree-Fock results. The Hartree-Fock anharmonic corrections agree well with the B3LYP corrections when calculated at the B3LYP geometry and from B3LYP normal coordinates, suggesting that the inclusion of electron correlation is not essential for the reliable calculation of cubic and quartic force constants.

Analytical calculations of frequency-dependent hypermagnetizabilities and Cotton-Mouton constants using London atomic orbitals.

We present the first gauge-origin-independent, frequency-dependent calculations of the hypermagnetizability anisotropy, which determines the temperature-independent contribution to magnetic-field-induced linear birefringence, the so-called Cotton-Mouton effect. A density-matrix-based scheme for analytical calculations of frequency-dependent molecular properties for self-consistent field models has recently been developed, which is also valid with frequency- and field-dependent basis sets. Applying this scheme to Hartree-Fock wave functions and using London atomic orbitals in order to obtain gauge-origin-independent results, we have calculated the hypermagnetizability anisotropy. Our results show that the use of London orbitals leads to somewhat better basis-set convergence for the hypermagnetizability compared to conventional basis sets and that London orbitals are mandatory in order to obtain reliable magnetizability anisotropies.

Analytic ab initio calculations of coherent anti-Stokes Raman scattering (CARS)

We present a theory for the analytic calculation of frequency-dependent polarizability gradients, and apply the methodology to the calculation of coherent anti-Stokes Raman scattering (CARS). The formalism used is based on an open-ended theory for the calculation of frequency-dependent molecular response properties of arbitrary order, also including contributions from perturbation-dependent basis sets. An important feature of our approach is the close connection between the formalism--which is fully matrix-based in an atomic orbital basis--and the implementation, allowing for the rapid implementation of higher-order molecular properties. Care is taken to allow the formalism to be utilized with linearly-scaling Hartree-Fock and density-functional theory codes. By avoiding the evaluation of responses due to geometry distortions, only 9 response equations need to be solved for the calculation of the CARS intensities, independent of the size of the molecular system. The theory is illustrated by calculations on a set of polyaromatic hydrocarbons using a DFT/B3LYP force field and Hartree-Fock polarizability gradients. Good agreement with the experimental CARS spectra of these compounds is obtained.