Wave Function Relaxation Contributions

In the final step of the EDA, the wave function of the molecule relaxes to its optimal form yielding the orbital interaction term AEoa. This term can be further partitioned into contributions by the orbitals belonging to different irreducible representations of the point group of the interacting system. Thus, it is possible to give energy contributions of the a and tt bonding contributions to a bond that has a mirror plane. More details about the EDA method in the framework of DFT can be found in a recent review article. ... 

The A -shell x-ray emission rates of molecules have been calculated with the DV-Xa method. The x-ray transition probabilites are evaluated in the dipole approximation by the DV-integration method using molecular wave functions. The validity of the DV-integration method is tested. The calculated values in the relaxed-orbital approximation are compared with those of the frozen-orbital approximation and the transition-state method. The contributions from the interatomic transitions are estimated. The chemical effect on the KP/Ka ratios for 3d elements is calculated and compared with the experimental data. The excitation mode dependence on the Kp/Ka ratios for 3d elements is discussed. 

The hcc s obtained at the B3LYP/EPR-II level are shown in Table 12. The calculated hcc s can be dissected into three terms a contribution due to the electronic and structural configurations assumed by the radicals in the gas phase (first column in Table 12) a contribution due to the solvent-induced polarization on the solute wave function without allowing any relaxation of the gas-phase geometry (direct solvent effect, second coliunn in Table 12), and a last contribution due to the solvent-induced geometry relaxation (indirect solvent effect, third column in Table 12). 

In most of the cases, an ultrasonic wave propagates adiabatically, so the (20) looks more naturally its right-hand side represents the adiabatic (non-relaxed) modulus and non-adiabatic contribution to the dynamic modulus. Recall that the relaxed (or isothermal) modulus should be regarded as quasi-static one. Figure 1 shows the frequency-dependent factor of non-adiabatic contribution as function of cox. One can see that transformation from isothermal-like to adiabatic-like propagation occurs in the vicinity cox = 1. The velocity of ultrasound is increased in this region, while the attenuation reaches its maximum value. 

The first contribution depends upon the second-order behavior of the Hamiltonian operator and the unperturbed reference state wave function, while the second term (which will be subsequently be referred to as the relaxation contribution or relaxation term ) depends on the derivative of the wave function. This is perhaps most easily appreciated by inserting the equation for the wave-function derivative into that for the second derivative of the energy, giving... 

Errors (/>(calc)-/ (expt)) in the relaxed crystal structures using a fixed repulsion-dispersion potential (as in Table 11.2) and various electrostatic models derived from a DMA of a 6-31G SCF wave function DMA, full multipoles up to hexadecapole SDMA, all multipoles scaled by 0.9 CHAR, just the charge component of the DMA. The r.m.s. % errors were calculated over the three cell lengths. The electrostatic contribution, (7estat, to the total lattice energy, Us, is given at both the experimental (for the SDMA model) and relaxed crystal structures. This can be compared with the experimental heat of sublimation Af/sutl (Chickos 1987), where available. 

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