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The state-averaged (SA) spin restricted ensemble referenced Kohn-Sham(REKS) method and its state interaction (SI) extension, SI-SA-REKS, enable one to describe correctly the shape of the ground and excited potential energy surfaces of molecules undergoing bond breaking/bond formation reactions including features such as conical intersections crucial for theoretical modeling of non-adiabatic reactions. Until recently, application of the SA-REKS and SI-SA-REKS methods to modeling the dynamics of such reactions was obstructed due to the lack of the analytical energy derivatives. In this work, the analytical derivatives of the individual SA-REKS and SI-SA-REKS energies are derived. The final analytic gradient expressions are formulated entirely in terms of traces of matrix products and are presented in the form convenient for implementation in the traditional quantum chemical codes employing basis set expansions of the molecular orbitals. The implementation and benchmarking of the derived formalism will be described in a subsequent article of this series. [ABSTRACT FROM AUTHOR]

Analytic second derivatives of the relativistic energy for the calculation of electric response properties are derived utilizing the normalized elimination of the small component (NESC) method. Explicit formulas are given for electric static dipole polarizabilities and infrared intensities by starting at the NESC representation of electric dipole moments. The analytic derivatives are implemented in an existing NESC program and applied to calculate dipole moments, polarizabilities, and the infrared spectra of gold- and mercury-containing molecules as well as some actinide molecules. Comparison with experiment reveals the accuracy of NESC second order electric response properties. [ABSTRACT FROM AUTHOR]

The analytical energy gradient of the normalized elimination of the small component (NESC) method is derived for the first time and implemented for the routine calculation of NESC geometries and other first order molecular properties. Essential for the derivation is the correct calculation of the transformation matrix U relating the small component to the pseudolarge component of the wavefunction. The exact form of ∂U/∂λ is derived and its contribution to the analytical energy gradient is investigated. The influence of a finite nucleus model and that of the picture change is determined. Different ways of speeding up the calculation of the NESC gradient are tested. It is shown that first order properties can routinely be calculated in combination with Hartree-Fock, density functional theory (DFT), coupled cluster theory, or any electron correlation corrected quantum chemical method, provided the NESC Hamiltonian is determined in an efficient, but nevertheless accurate way. The general applicability of the analytical NESC gradient is demonstrated by benchmark calculations for NESC/CCSD (coupled cluster with all single and double excitation) and NESC/DFT involving up to 800 basis functions. [ABSTRACT FROM AUTHOR]

Analytic expressions are derived for the evaluation of derivatives of the total molecular energy with respect to external parameters (nuclear coordinates, external electric fields, etc.) within the relativistic regular approximation. The presented formalism employs the spectral resolution of the identity avoiding, however, the explicit use of an auxiliary basis set in the calculation of the matrix elements of the regular relativistic Hamiltonian. The final formulas for the total energy and energy derivatives are presented in matrix form suitable for implementation into standard quantum chemical packages. Results of benchmark calculations for gold containing diatomic molecules and for xenone hexafluoride performed at the Hartree-Fock and various correlation corrected levels of theory are presented and discussed. [ABSTRACT FROM AUTHOR]

By expanding the Foldy-Wouthuysen representation of the Dirac equation near the free- particle solution it is shown that the Hamiltonian of the zeroth-order regular approximation (ZORA) leads to an infinite summation of the leading relativistic corrections to the free- particle, non-relativistic energy. The analysis of the perturbation expansion of the ZORA Hamiltonian reveals that the ZORA Hamiltonian recovers all terms of the Breit-Pauli theory to second order. This result is general and applies not only to hydrogen-like atomic ions (as was demonstrated before) but also to a wide variety of physical problems. ZORA is analogous to the random phase approximation in many-body theory in the sense that both methods include an infinite-order summation of the asymptotically non-vanishing terms. This highlights the difference between ZORA and the Douglas-Kroll method, with the latter being analogous to finite-order many-body perturbation theory. On the basis of this analysis the performance of ZORA when calculating various molecular properties is discussed. [ABSTRACT FROM AUTHOR]