Atomic orbital basis function derivatives

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

In these and subsequent expressions are used to denote molecular orbitals whereas p,q,r,s are atomic orbitals (basis functions). Thus the quantity e" is formed in the AO basis and transformed into the MO basis. This quantity has the same structure as a Fock operator but is constructed using the derivatives h and of the one-electron and two-electron integrals. A compact notation for this operation is... 

There is danger of error only when such elements are used exclusively.) The atomic orbital basis consists of a p function on each of the four carbon atoms Figure 11.15 illustrates these orbitals and the derived symmetry correct orbitals. As a result of the reflection plane that carries one ethylene into the other, these molecular orbitals are delocalized over both molecules. Figure 11.15 also shows the orbitals of the product we consider only the two C—C bonds formed and ignore the other two, which were present from the beginning and did not undergo any change. 

At the HF-SCF level of theory, the wave function is determined completely by the molecular orbitals. In the vast majority of cases, these are given by linear combinations of atom-centered basis functions, and these MO coefficients are obtained by the self-consistent field procedure. The first-order change to the wave function is therefore governed by the first-order change in the MO coefficients. It is not difficult to work out expressions for the derivatives of the MO coefficients,22 and one obtains... 

Hence, if we are to make bE = 0, we can avoid these terms. But to do so we have to have E optimum with respect to the location of the atomic basis functions, t (R) the MO coefficients, c(R) and the Cl coefficients, C(R). The first cannot be satisfied unless the atomic orbital basis set is floated off the atomic centers to an optimum location [105], while the second requires optimum MO coefficients, and the third optimum Cl coefficients. In practice, we will introduce atomic orbital derivatives explicitly, so the AOs can follow their atoms. Now focusing only on the MO and Cl coefficients, in SCF we have optimum MOs and no Cl term. In MCSCF, both terms would vanish, whUe in Cl, the MO derivatives would remain, but the Cl coefficients contribution would vanish. In the non-variational coupled-cluster theory, neither will vanish and this means that CC theory forces us into some new considerations for analytical forces. 

The electronic energy W in the Bom-Oppenlieimer approxunation can be written as W= fV(q, p), where q is the vector of nuclear coordinates and the vector p contains the parameters of the electronic wavefimction. The latter are usually orbital coefficients, configuration amplitudes and occasionally nonlinear basis fiinction parameters, e.g., atomic orbital positions and exponents. The electronic coordinates have been integrated out and do not appear in W. Optimizing the electronic parameters leaves a function depending on the nuclear coordinates only, E = (q). We will assume that both W q, p) and (q) and their first derivatives are continuous fimctions of the variables q- and py... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

In general, all 17 s-primitives contribute to each s-derived molecular orbital. Obviously, the tighter Gaussians will contribute more strongly to the inner-shell molecular orbitals and the more diffuse Gaussians to the valence i-orbitals. Nevertheless, it is impossible (and also not desired) to make a connection between basis functions and atomic orbitals. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

This localization scheme permits tire assignment of hybridization both to the atomic lone pairs and to each atom s contributions to its bond orbitals. Hybridization is a widely employed and generally useful chemical concept even though it has no formal basis in the absence of high-syrnmetry constraints. Witli NBO analysis, tire percent s and p character (and d, f, etc.) is immediately evident from tire coefficients of tire AO basis functions from which the NAO or NBO is formed. In addition, population analysis can be carried out using the NBOs to derive partial atomic charges (NPA, see Section 9.1.3.2). 

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