Kinetic-energy integrals

At this point it might be helpful to summarize what has been done so far in terms of effective potentials. To obtain the QFH correction, we started with an exact path integral expression and obtained the effective potential by making a first-order cumulant expansion of the Boltzmann factor and analytically performing all of the Gaussian kinetic energy integrals. Once the first-order cumulant approximation is made, the rest of the derivation is exact up to (11.26). A second-order expansion of the potential then leads to the QFH approximation. 

The partition function can be written as the product of K, the kinetic energy integral, and Z, the configurational integral,... 

In all cases considered, differences in free energy for a given material in two different states or derivatives of the free energy are calculated and the contributions of the kinetic energy integrals to these quantities will exactly cancel. Since the results are not altered by its omission, for the sake of brevity, we shall not consider the kinetic energy integral. 

Calculations show that this kinetic energy integral is, for example, negative for a bonded pair, 0X, 02 and does approximate the value of H12 rather well (12).. 

The kinetic energy integral follows easily from the overlap integral. We write the kinetic energy operator... 

SERGIY BUBIN, MAURICIO CAFIERO AND LUDWIK ADAMOWICZ The kinetic energy integral is given by... 

The Fock integrals first encountered in equation (A.45) are constructed from kinetic energy integrals, nuclear-electron attraction integrals, and two-electron repulsion integrals, as follows, continuing from equation (A.49) ... 

The kinetic energy integrals are collected as the matrix T, whose elements are defined by... 

Here //co,c(l) has been dissected into a kinetic energy integral T and two potential energy integrals, F(H) and F(He). From the definition of the operator //co,c (Eq. 5.64 = 5.19) and the Roothaan-Hall expression for the integral //core (Eq. 5.79) we see that (the (1) emphasizes that these integrals involve the coordinates of only one electron) ... 

This formula expresses the surface integral obtained by integrating the kinetic energy integral by parts for open-channel orbital functions of the specified asymptotic form. In matrix notation,... 

Using this result after some transformations the derivative of the kinetic energy integral becomes ... 

Cusachs reported (4) that the repulsive terms in the W-H model which assumes that electron repulsion and nuclear repulsion cancel nuclear-electron attraction, consist of one-electron antibonding terms only. Cusachs noted Ruedenberg s observation that the two-center kinetic energy integral is proportional to the square of the overlap integral rather than the first power. Cusachs used this to develop the approximation ... 

Fluctuating components of gas velocity are selected from a Gaussian distribution with variance derived from the local value of turbulent kinetic energy. Integration of the particle equation of motion yields the particle position at any given instant in time. 

Note the appearance of the kinetic energy integral in Eq. [19c], The last term of Eq. [19d] results from invoking the Mulliken integral approximation, Eq. [17a],... 

Here, V, iff, and (X, Y = L or S, L = S, and S = L) are kinetic energy integral, electron-nuclear attraction integral, exchange-correlation potential. 

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