Kinetic-energy integrals integral evaluation

Note that, in (9.6.26), the abscissae and weights are the same for all integrals of the same quantum number i + j + e. The Gauss-Hermite scheme for multipole-moment integrals may easily be extended to the evaluation of kinetic-energy integrals. 

The P matrix involves the HF-LCAO coefficients and the hi matrix has elements that consist of the one-electron integrals (kinetic energy and nuclear attraction) over the basis functions Xi - Xn - " h matrix contains two-electron integrals and elements of the P matrix. If we differentiate with respect to parameter a which could be a nuclear coordinate or a component of an applied electric field, then we have to evaluate terms such as... 

The obstacle to simultaneous quantum chemistry and quantum nuclear dynamics is apparent in Eqs. (2.16a)-(2.16c). At each time step, the propagation of the complex coefficients, Eq. (2.11), requires the calculation of diagonal and off-diagonal matrix elements of the Hamiltonian. These matrix elements are to be calculated for each pair of nuclear basis functions. In the case of ab initio quantum dynamics, the potential energy surfaces are known only locally, and therefore the calculation of these matrix elements (even for a single pair of basis functions) poses a numerical difficulty, and severe approximations have to be made. These approximations are discussed in detail in Section II.D. In the case of analytic PESs it is sometimes possible to evaluate these multidimensional integrals analytically. In either case (analytic or ab initio) the matrix elements of the nuclear kinetic energy... 

Note that the kinetic energy of the flow has been neglected. So far this result is the same as for the constant-area channel with no surface chemistry (Eq. 16.29). Using the definition of enthalpy and evaluating the integral, we have... 

Equation (4.15) is obtained by analytic evaluation of the u integration over the kinetic energy terms in Eq. (4.13). 

There have been two principal methods developed to evaluate the kinetic energy using path integral methods. One method, based on Eq. (3.5), has been termed the T-method and the other, based on Eq. (4.1), has bwn termed the //-method. In discretized path integral calculations the T-method and the //-method have similar properties, but in the Fourier method the expressions and the behavior of the kinetic energy evaluated by Monte Carlo techniques are different. 

For an oscillator potential, the variance obtained in Monte Carlo evaluations of Eq. (4.36) can be shownto be strictly independent of and our experience with other potentials has shown no appreciable dependence of the variance on Although the T-method expression for the kinetic energy is simpler and appears to require less numerical work, our experience with the H-method has been more favorable. The extra work in evaluating the onedimensional u integrations is more than compensated by the smaller number of Monte Carlo points required in the estimation of the averages. 

Our evaluation of the Fundamental Integral cannot proceed further than (18) unless we now specify the two-electron function f(x). We are now in a position to consider some of the integral types which arise in quantum chemical calculations overlap, kinetic-energy, electron-repulsion, nuclear-attraction and anti-coulomb. 

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