Perturbation-dependent Basis Sets Research Articles

Open-ended recursive calculation of single residues of response functions for perturbation-dependent basis sets.

We present theory, implementation, and applications of a recursive scheme for the calculation of single residues of response functions that can treat perturbations that affect the basis set. This scheme enables the calculation of nonlinear light absorption properties to arbitrary order for other perturbations than an electric field. We apply this scheme for the first treatment of two-photon circular dichroism (TPCD) using London orbitals at the Hartree-Fock level of theory. In general, TPCD calculations suffer from the problem of origin dependence, which has so far been solved by using the velocity gauge for the electric dipole operator. This work now enables comparison of results from London orbital and velocity gauge based TPCD calculations. We find that the results from the two approaches both exhibit strong basis set dependence but that they are very similar with respect to their basis set convergence.

A general, recursive, and open‐ended response code

We present a new implementation of a recent open-ended response theory formulation for time- and perturbation-dependent basis sets (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) at the Hartree-Fock and density functional levels of theory. A novel feature of the new implementation is the use of recursive programming techniques, making it possible to write highly compact code for the analytic calculation of any response property at any valid choice of rule for the order of perturbation at which to include perturbed density matrices. The formalism is expressed in terms of the density matrix in the atomic orbital basis, allowing the recursive scheme presented here to be used in linear-scaling formulations of response theory as well as with two- and four-component relativistic wave functions. To demonstrate the new code, we present calculations of the third geometrical derivatives of the frequency-dependent second hyperpolarizability for HSOH at the Hartree-Fock level of theory, a seventh-order energy derivative involving basis sets that are both time and perturbation dependent.

Hyper Raman spectra calculated in a time-dependent Hartree–Fock method

Hyper Raman scattering (HRS) of the benzonitrile (BN) and 1,3,5-trinitro-1,3,5-triazacyclohexane (RDX) molecules is studied by means of ab initio calculations. The computational procedure employs a recently developed methodology for the analytic calculations of frequency-dependent polarizability gradients of arbitrary order, including perturbation dependent basis sets. The result are compared to normal Raman scattering (NRS) and coherent anti-Stokes Raman scattering (CARS) that previously have been studied using the same technology. It is found that some suppressed or silent modes in CARS and NRS spectra are clearly seen in HRS, and that although under general excitation conditions the HRS intensities are much lower than for CARS and NRS, HRS provides complementary information useful for target identification.

Gauge-origin independent calculations of Jones birefringence

We present the first gauge-origin independent formulation of Jones birefringence at the Hartree-Fock level of theory. Gauge-origin independence is achieved through the use of London atomic orbitals. The implementation is based on a recently proposed atomic orbital-based response theory formulation that allows for the use of both time- and perturbation-dependent basis sets [Thorvaldsen, Ruud, Kristensen, Jo̸rgensen, and Coriani, J. Chem. Phys. 129, 214108 (2008)]. We present the detailed expressions for the response functions entering the Jones birefringence when London atomic orbitals are used. The implementation is tested on a set of polar and dipolar molecules at the Hartree-Fock level of theory. It is demonstrated that London orbitals lead to much improved basis-set convergence, and that the use of small, conventional basis sets may lead to the wrong sign for the calculated birefringence. For large basis sets, London orbitals and conventional basis sets converge to the same results.

The ab initio calculation of molecular electric, magnetic and geometric properties

We give an account of some recent advances in the development of ab initio methods for the calculation of molecular response properties, involving electric, magnetic, and geometric perturbations. Particular attention is given to properties in which the basis functions depend explicitly both on time and on the applied perturbations such as perturbations involving nuclear displacements or external magnetic fields when London atomic orbitals are used. We summarize a general framework based on the quasienergy for the calculation of arbitrary-order molecular properties using the elements of the density matrix in the atomic-orbital basis as the basic variables. We demonstrate that the necessary perturbed density matrices of arbitrary order can be determined from a set of linear equations that have the same formal structure as the set of linear equations encountered when determining the linear response equations (or time-dependent self-consistent-field equations). Additional components needed to calculate properties involving perturbation-dependent basis sets are flexible one- and two-electron integral techniques for geometric or magnetic-field differentiated integrals; in Kohn-Sham density-functional theory (KS-DFT), we also need to calculate derivatives of the exchange-correlation functional. We describe a recent proposal for evaluating these contributions based on automatic differentiation. Within this framework, it is now possible to calculate any molecular property for an arbitrary self-consistent-field reference state, including two- and four-component relativistic self-consistent-field wave functions. Examples of calculations that can be performed with this formulation are presented.

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.

Reduced-size polarized basis sets for calculations of molecular electric properties. IV. First-row transition metals

Recent studies of the perturbation-dependent basis sets have indicated the possibility of a significant reduction of the size of the usual CGTO sets without considerable loss of accuracy in calculations of molecular electric properties. The resulting (ZPolX) basis sets have been developed for several atoms of the first and second row of the Periodic Table. The same method of the ZPolX basis set generation is extended for the first-row transition metals and the corresponding contracted ZPolX basis sets of the size [6s5p3d1f] are determined for both nonrelativistic and scalar relativistic calculations. The performance of the ZPolX basis sets is verified in calculations on the first-row transition metal oxides at the level of the ROHF, ROHF/CASPT2, and ROHF/CCSD(T) approximations. Also the study of the dipole polarizability of TiCl4 confirms the excellent features of these very compact basis sets. The ZPolX basis sets for nonrelativistic and relativistic calculations of molecular electric properties are available on the web page http://www.chem.uni.torun.pl/zchk/basis-sets.html.

Geometrical derivatives and magnetic properties in atomic-orbital density-based Hartree–Fock theory

A reformulation of Hartree–Fock theory for time-independent molecular properties with perturbation-dependent basis sets and which refers strictly to the atomic-orbital basis is presented. The formalism is based on a recently proposed exponential parametrization of the one-electron atomic-orbital density matrix. In the presented formulation, only multiplications and additions of sparse matrices are needed. Linear scaling with system size is therefore obtainable, making this formulation ideally suited to large molecular systems. The paper contains general formulas for molecular energy derivatives up to fourth order, with special attention given to molecular gradients, molecular Hessians, magnetizabilities, and nuclear magnetic shieldings.

The effect of correlation on molecular magnetizabilities and rotational g tensors

Using multiconfigurational self-consistent field (MCSCF) wave functions and perturbation-dependent basis sets, the effect of electron correlation on molecular magnetizabilities and rotational g tensors is investigated. The eight molecular systems considered (H2O, NH3, HF, C2H2, CO, H2CO, O3, and LiH) vary in the importance and relative magnitudes of the static and dynamic correlation contributions. The results for O3 are the first correlated calculations of the rotational g tensor of this system. We confirm previous findings that, except for systems with large static correlation effects, the effect of correlation on molecular magnetizabilities is small. A somewhat larger correlation contribution is usually observed for the rotational g tensor, although this property is also rather insensitive to the correlation treatment. Agreement with experimental rotational g tensors is only fair and estimates of rovibrational corrections are needed to assess properly the accuracy of theoretically calculated rotational g tensors.

A numerically stable orbital connection for the calculation of analytical Hessians using perturbation-dependent basis sets

It is demonstrated that calculations of second-order molecular properties using perturbation-dependent basis sets are most efficiently carried out using the natural connection.

Orbital connections for perturbation-dependent basis sets

The use of perturbation-dependent basis sets is analysed with emphasis on the connection between the basis sets at different values of the perturbation strength. A particular connection, the natural connection, that minimizes the change of the basis set orbitals is devised and the second quantization realization of this connection is introduced. It is shown that the natural connection is important for the efficient evaluation of molecular properties and for the physical interpretation of the terms entering the calculated properties. For example, in molecular Hessian calculations the natural connection reduces the size of the relaxation term, leading to faster convergence of the response equations. The physical separation of the terms also means that first-order non-adiabatic coupling matrix elements can be obtained in a very simple way from a molecular Hessian calculation.

Orbital connections for perturbation-dependent basis sets

Orbital connections for perturbation-dependent basis sets