Analytic evaluation of the dipole Hessian matrix in coupled-cluster theory

Abstract

The general theory required for the calculation of analytic third energy derivatives at the coupledcluster level of theory is presented and connected to preceding special formulations for hyperpolarizabilities and polarizability gradients. Based on our theory, we have implemented a scheme for calculating the dipole Hessian matrix in a fully analytical manner within the coupled-cluster singles and doubles approximation. The dipole Hessian matrix is the second geometrical derivative of the dipole moment and thus a third derivative of the energy. It plays a crucial role in IR spectroscopy when taking into account anharmonic effects and is also essential for computing vibrational corrections to dipole moments. The superior accuracy of the analytic evaluation of third energy derivatives as compared to numerical differentiation schemes is demonstrated in some pilot calculations.

Analytic calculations of anharmonic infrared and Raman vibrational spectra

“…We have also shown that evaluating the energy and property derivatives by numerical differentiation is prone to numerical instabilities, as also noted elsewhere, 51 so that obtaining reliable numerical derivatives can prove difficult for general molecular systems, and we have seen that the errors thus introduced can significantly affect the calculated results, whereas analytic approaches would be free of these sources of error. Because of this, we believe that analytical derivatives of high order is an important step in making the inclusion of anharmonic corrections in calculated infrared and Raman spectra routine, leading to an improved understanding of the importance and occurrence of anharmonic effects in vibrational spectroscopies.…”

“…48 For the IR and (regular) Raman spectroscopies, programs that allow for the analytic calculation of the required first-order geometric derivatives of the dipole moment and polarizability, respectively, have been available for some time. [49][50][51] The calculation of anharmonic corrections to the intensities in these spectroscopies requires both the development of the necessary vibrational perturbation theory 38 to obtain expressions for these corrections and the possibility of calculating second-and third-order geometric polarization property derivatives, as well as the cubic and quartic force constants, that enter into these expressions. Programs that would allow for the analytic calculation of some of these properties are available, but such calculations have mainly been restricted to the HF level of theory, and for some of the properties (and more so if a DFT description is desired), the researcher has had to resort to numerical differentiation.…”

Geometric Energy Derivatives at the Complete Basis Set Limit: Application to the Equilibrium Structure and Molecular Force Field of Formaldehyde

“…Although the numerical gradients used here benefit from a clearly parallelized computational strategy, analytic gradients will always be preferred when available as they are much faster to compute, are independent of system size, and suffer much less from numerical errors. , A brief derivation of how common CBS formulas may be extended to include first-order geometric derivatives is presented below. Solution of the a , b , c coefficients in an exponential fit produces an easily implemented form of the CBS extrapolation of Hartree–Fock energies by way of In the above form, Hartree–Fock energies computed using correlation-consistent basis sets are presented as E n where the highest n corresponds to the largest basis set used.…”

“…The accuracy of the energy levels directly involved in the triad benefits from an effective Hamiltonian treatment (as shown in eq 19) following deperturbation, but x 25 * , x 35 * , and x 65 * and their corresponding unmixed energy levels do not. Errors are over 40 cm −1 for CCSDT(Q)/CBS VPT2 predictions of 3 1 5 1 (b 2 ) = 4289.28 cm −1 and 5 1 6 1 (a 1 ) = 4042.24 cm −1 relative to the gas-phase observations of 4335.09709 (60) and 4083.1(10) cm −1 , respectively. 29,124 Even the errors for the eigenvalues of H eff are far too high relative to the rest of the data set.…”

Analytic energy derivatives in relativistic quantum chemistry

“…The advantage of the analytic scheme is therefore obvious, in particular with increasing system size. The computational efficiency of analytic schemes is even more prominent for computing higher order derivatives; the scaling behaviors of analytic schemes for evaluating n th order geometrical derivatives of the energy are and for odd and even derivatives, respectively, where denotes the number of perturbation parameters. This should be compared with the scaling of the corresponding numerical schemes.…”