Geometric derivation

J0rgensen P and Simons J (eds) 1986 Geometrical Derivatives of Energy Surfaces and Molecular Properties (Boston, MA Reidel)... 

While LAO/GIAO had been proposed well before the advent of modem computational chemistry, it was only developments in calculating (geometrical) derivatives of the energy (and wave function) that made it practical to use field-dependent orbitals. ... 

Whitaker, S, A Simple Geometrical Derivation of the Spatial Averaging Theorem, Chemical Engineering Education Winter, 18, 1985. 

The purely geometrical derivatives of W(N, Q) include the forces acting on nuclei along the internal geometric coordinates... 

The four blocks of V can be alternatively expressed in terms of the principal geometric derivatives defining the generalized Hessian of Equation 30.8. This can be accomplished first by expressing AQ as function of AN and AF, using the second Equation 30.9, and then by inserting the result into the first Equation 30.9 ... 

We infer that final products, C, D and E, stem from the same metal-carbon a-bonded intermediate B, and their relative amounts are due to kinetic factors. Carrying out the same reaction starting from an E geometric derivative, the EZ isomerization is, as... 

The geometrical derivatives of the PCM-TDDFT excitation energy of a given excited state can be used to obtain the equilibrium geometry of that state. From this equilibrium geometry the excited state can reach the ground state by a vertical emission process whose emission energy can be determined by a proper application of the non-equilibrium scheme presented in the previous section. 

Analytical Calculation of Geometrical Derivatives in Molecular Electronic Structure Theory... 

By combining the order-by-order expansions of the Hamiltonian and the wave function we identify the geometrical derivatives Wcomputer implementations of these expressions. 

In Section VII we describe how expressions for geometrical derivatives of molecular properties may be derived using the formalism developed for energy derivatives. We also discuss alternative definitions that may be used to determine geometrical derivatives of molecular properties for wave functions which do not satisfy the Hellmann-Feynman relationship for the property in question. Finally, in Section VIII we describe how translational and rotational symmetries may be used to reduce the cost of derivative calculations. 

Inserting the power-series expansion of A [Eq. (63)] in the total energy expression in Eq. (55) allows us to identify the geometrical derivatives for the MCSCF wave function as... 

The geometrical derivatives may be identified by inserting the power series expansion of P [Eq. (113)] into the Cl energy expression [Eq. (111)]. We obtain... 

The left-hand side of Eq. (177) has a structure similar to the electronic gradient vector in variational wave function calculations. Unlike variational calculations, Eq. (177) cannot be used to determine the response parameters t(n) in a CC calculation. However, for the calculation of the nth geometrical derivative W n Eq. (177) eliminates r(n), which would otherwise appear in the calculation. In fact, we show below that the calculation of (3N-6)n response amplitudes t(n) is replaced by the solution of one set of linear equations of similar but simpler structure. By inserting Eq. (176) in Eq. (177) and rearranging terms, we see that X fulfills the equations... 

In this section we discuss the analytical calculation of geometrical derivatives of molecular properties. The emphasis is on properties related to variations in the electric field. 

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