Intermolecular perturbation second-order energy

The first-order energy vanishes in the present case since the geometry of the fragment A, and that of the fragment B as well, is assumed to be unaffected by intermolecular perturbation. Note that the first-order orbital mixing between and (i.e., f(l ) leads to the second-order energy change The second-... 

Similar to the distributed-multipole expansion of molecular electrostatic fields, one can derive a distributed-polarizability expansion of the molecular field response. We can start by including the multipole-expansion in the perturbing Hamiltonian term W = Qf(p, where we again use the Einstein sum convention for both superscripts a, referencing an expansion site, and subscripts t, which summarize the multipole components (/, k) in just one index. Using this approximation for the intermolecular electrostatic interaction, the second-order energy correction now reads ... 

Empirically corrected DFT theories almost invariably go back to second-order perturbation theory with expansion of the interaction Hamiltonian in inverse powers of the intermolecular distance, leading to R 6, R x, and R 10 corrections to the energy in an isotropic treatment (odd powers appear if anisotropy is taken into account [86]). 

In intermolecular perturbation theory one of the major problems concerns electron exchange between molecules. In the method described here exchange is limited to single electrons. This simplification is definitely a good approximation at large intermolecular distances. The energy of interaction between the molecules, AE (R), is obtained as a sum of first order, second order, and higher order contributions ... 

Fig. 2. Second-order contributions to intermolecular perturbation energies (schematic description by orbital excitations within the framework of the independent particle model AEpol and zI-Echt are represented by single excitations, AEms by correlated double excitations)...

The last important contribution to intermolecular energies that will be mentioned here, the dispersion energy (dEnis). is not accessible in H. F. calculations. In our simplified picture of second-order effects in the perturbation theory (Fig. 2), d mS was obtained by correlated double excitations in both subsystems A and B, for which a variational wave function consisting of a single Slater determinant cannot account. An empirical estimate of the dispersion energy in Li+...OH2 based upon the well-known London formula (see e.g. 107)) gave a... 

The most convenient procedure for attaining the minimum of the second order perturbation expression of the energy, so as to generate the optimized virtual orbitals, depends on the kind of problem being studied. In the case of intermolecular interactions, convergence is quite easy with just a gradient-based procedure. The minimization scheme can be recast in such a way that the coefficients of the improved virtual orbitals can be obtained, at each step, by a resolution of a linear system of NA+NB equations. Specifically. 

Chalasinski and Szczesniak have provided a means of decomposing the correlation contribution to the interaction energy into four separate terms. Their philosophy takes the electron exchange operator as a second perturbation in the spirit of many-body perturbation theory, with molecular interaction as the first perturbation in their intermolecular Mpller-Plesset perturbation theory (IMPPT). At the level of second order of the correlation operator, they obtain a number of separate terms. The first is the dispersion energy, e... 

Since the symmetry-adapted perturbation theory provides the basis for the understanding of weak intermolecular interactions, it is useful to discuss the convergence properties of the sapt expansion. High-order calculations performed for model one-electron (Hj) (30), two-electron (H2) (14, 15), and four-electron (He and He-Hz) (31) systems show that the sapt series converges rapidly. In fact, already the second-order calculation reproduces the exact variational interaction energies with errors smaller than 4%. Several recent applications strongly indicate that this optimistic result holds for larger systems as well. 

Electron correlation effects are known to be impx>rtant in systems with weak interactions. Studies of van der Waals interetctions have established the importance of using methods which scale linearly with the number of electrons[28] [29]. Of these methods, low-order many-body perturbation theory, in particular, second order theory, oflfers computational tractability combined with the ability to recover a significant firaction of the electron correlation energy. In the present work, second order many-body perturbation theory is used to account for correlation effects. Low order many-body perturbation theory has been used in accurate studies of intermolecular hydrogen bonding (see, for example, the work of Xantheas and Dunning[30]). 

The exact nature of the intermolecular forces need not be specified for the development of the theory. However, in their original presentation Maier and Saupe assumed that the stability of the nematic phase arises from the dipole-dipole part of the anisotropic dispersion forces. The second order perturbed energy of the Coulomb interaction between a pair of molecules 1 and 2 is given by... 

The first attempt to develop a statistical model of the cholesteric phase was by Goossens who extended the Maier-Saupe theory to take into account the chiral nature of the intermolecular coupling and showed that the second order perturbation energy due to the dipole-quadrupolar interaction must be included to explain the helicity. However, a diflUculty with this and some of the other models that have since been proposed is that in their present form they do not give a satisfactory explanation of the fact that in most cholesterics the pitch decreases with rise of temperature. 

To second order in perturbation theory the change in energy arising from the intermolecular interactions is... 

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