Quantum corrections Wigner-Kirkwood

Classical and semiclassical moment expressions. The expressions for the spectral moments can be made classical by substituting the classical distribution function, g(R) = exp (— V(R)/kT), for the quantum expressions. Wigner-Kirkwood corrections are known which account to lowest order for the static quantum corrections, Eq. 5.44 [177, 292]. For the second and higher moments, dynamic quantum corrections must also be made [177]. As was mentioned in the previous Chapter, such semiclassical corrections are useful in supplementing quantum computations of the spectral moments at large separations where the quantum effects are small the computational effort of quantum calculations, which is substantial at large separations, may thus be avoided. 

This is the semi-classical, second, binary moment. While the zeroth and first moments require only static quantum corrections of the Wigner-Kirkwood type, the second and all higher moments require also dynamical corrections involving yC4 and higher moments. 

Other Wigner-Kirkwood corrections. The functions gE(R) and gM R) introduced above for the computation of the second moment, Eq. 5.39, have also been given in semi-classical form [292], Since these are very useful to supplement quantum moment calculations at large separations R, we will simply quote here the results,... 

The Wigner-Kirkwood corrected results (starred lines in the Table) for the lowest temperature may be uncertain by some fraction of the quantum correction and should be considered rough estimates. 

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

Spectral moments can also be computed from classical expressions with Wigner-Kirkwood quantum corrections [177, 189, 317] of the order lV(H2). For the quadrupole-induced 0223 and 2023 components of H2-H2, at the temperature of 40 K, such results differ from the exact zeroth, first and second moments by -10%, -10%, and +30% respectively. For the leading overlap-induced 0221 and 2021 components, we get similarly +14%, +12%, and -56%. These numbers illustrate the significance of a quantum treatment of the hydrogen pair at low temperatures. At room temperature, the semiclassical and quantum moments of low order differ by a few percent at most. Quantum calculations of higher-order moments differ, however, more strongly from their classical counterparts. 

Table 6.3. Various computed zeroth and first moments M , with and without lowest-order Wigner-Kirkwood quantum corrections, for a hydrogen-argon mixture at 195 K. Units are 10-34 J amagat-N for the zeroth moment, and 10-21 W amagat N for the first moments, with N = 2 and 3 for binary and ternary moments, respectively. An asterisk means that Wigner-Kirkwood corrections were not made for the entries of that line. The superscripts 12 and 122 stand for H2-Ar and H2-Ar-Ar, respectively [296].

In recognition of the quantum nature of the intermolecular O H hydrogen bond, interest exists in investigating ways of including a quantum mechanical correction of the interaction potential and its effect on solvation. The numerous techniques for handling quantum effects in water simulations include explicit wavefunction techniques, the Wigner-Kirkwood expansion of the classical potential surface, and approximate corrections to... 

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