Polarization perturbation theory

M0ller-Plesset Perturbation Theory Polarization Propagator... 

Mi ller-Plesset Perturbation Theory Polarization Propagator 213 The matrix form of the polarization propagator, (3.159), can thus be written as... 

M0ller-Plesset Perturbation Theory Polarization Propagator 215 where the elements of the RPA and matrices and of the... 

M0ller—Plesset Perturbation Theory Polarization Propagator 223... 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

These are the analogous equations to the response equations for Mpller Plesset perturbation theory polarization propagators or MCSCF linear response functions in Eqs. (10.29) and (11.46). However, there are a few important differences. First, in... 

Perturbation theory is a natural tool for the description of intemioleciilar forces because they are relatively weak. If the interactmg molecules (A and B) are far enough apart, then the theory becomes relatively simple because tlie overlap between the wavefiinctions of the two molecules can be neglected. This is called the polarization approximation. Such a theory was first fomuilated by London [3, 4], and then refomuilated by several others [5, 6 and 7]. 

The perturbation theory described in section Al.5.2,1 fails completely at short range. One reason for the failure is that the multipole expansion breaks down, but this is not a fiindamental limitation because it is feasible to construct a non-expanded , long-range, perturbation theory which does not use the multipole expansion [6], A more profound reason for the failure is that the polarization approximation of zero overlap is no longer valid at short range. 

Adams W H 1994 The polarization approximation and the Amos-Musher intermolecular perturbation theories compared to infinite order at finite separation Chem. Phys. Lett. 229 472... 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

The effeet of adding in the py orbitals is to polarize the 2s orbital along the y-axis. The amplitudes Cn are determined via the equations of perturbation theory developed below the ehange in the energy of the 2s orbital eaused by the applieation of the field is expressed in terms of the Cn eoeffieients and the (unperturbed) energies of the 2s and npy orbitals. 

The concerns we have expressed are bound to get even more acute if the problem under study demands that we are able to adequately describe distortion effects induced in the electron distribution by external fields. The evaluation of linear (and, still more, non linear) response funetions [1] by perturbation theory then forces one to take care also of the nonoccupied portion of the complete orbital spectrum, which is entrusted with the role of representing the polarization caused by the external fields in the unperturbed electron distribution [4], ... 

Details of the extended triple zeta basis set used can be found in previous papers [7,8]. It contains 86 cartesian Gaussian functions with several d- and f-type polarisation functions and s,p diffuse functions. All cartesian components of the d- and f-type polarization functions were used. Cl wave functions were obtained with the MELDF suite of programs [9]. Second order perturbation theory was employed to select the most energetically double excitations, since these are typically too numerous to otherwise handle. All single excitations, which are known to be important for describing certain one-electron properties, were automatically included. Excitations were permitted among all electrons and the full range of virtuals. 

Orr BJ, Ward JF (1971) Perturbation theory of the non-linear optical polarization of an isolated system. Mol Phys 20 513-526... 

In a perturbation theory treatment of the total (not just electrostatic) interaction between the molecule and the point charge, QV(r) is the first-order term in the expression for the total interaction energy (which would include polarization and other effects). 

When a radical is oriented such that the magnetic field direction is located by the polar and azimuthal angles, 6 and cp, relative to the g-matrix principal axes, the resonant field is given, to first order in perturbation theory, by 4... 

From the viewpoint of quantum mechanics, the polarization process cannot be continuous, but must involve a quantized transition from one state to another. Also, the transition must involve a change in the shape of the initial spherical charge distribution to an elongated shape (ellipsoidal). Thus an s-type wave function must become a p-type (or higher order) function. This requires an excitation energy call it A. Straightforward perturbation theory, applied to the Schroedinger aquation, then yields a simple expression for the polarizability (Atkins and Friedman, 1997) ... 

As a further illustration of the dependence of n i 7t pi-backbonding interactions on metal and ligand character, we may compare simple NiL complexes of nickel with carbonyl (CO), cyanide (CN-), and isocyanide (NC-) ligands, as shown in Fig. 4.41. This figure shows that the nNi 7rL pi-backbonding interaction decreases appreciably (from 28.5 kcal mol-1 in NiCO to 6.3 kcalmol-1 in NiNC-, estimated by second-order perturbation theory) as the polarity of the 7Tl acceptor shifts unfavorably away from the metal donor orbital. The interaction in NiCO is stronger than that in NiCN- partially due to the shorter Ni—C distance in the... 

The values associated with the contours in Fig. 8.1 correspond to the interaction energies of a proton with the unperturbed charge distribution of the molecule. It must, of course, be recognized that the latter will not remain unperturbed as the proton approaches. (There have been several attempts to take such polarization effects into account, for instance by means of perturbation theory [7, 10, 27, 28].) Nevertheless, the Vmin can be quite effective in ranking protonation sites if these are chemically similar, for example the nitrogens in a series of azines [8, 29, 30]. Problems can arise, however, when the charge-transfer capabilities of the sites inherently differ significantly, e.g. NH3 compared with PH3 [31, 32]. 

As reviewed above, when a solute is placed in a dielectric medium, it electrically polarizes that medium. The polarized medium produces a local electrostatic field at the site of the solute, this field polarizes the solute, and the polarized solute interacts with the polarized medium. The interaction is typically too large to be treated by perturbation theory, and some sort of self-consistent treatment of polarized solute and polarized medium is more appropriate. At this point several options present themselves. It promotes orderly discussion to classify these... 

In summary, density functional theory provides a natural framework to discuss solvent effects in the context of RF theory. A general expression giving the insertion energy of an atom or molecule into a polarizable medium was derived. This expression given in Eq (83), when treated within a first order perturbation theory approach (i.e. when the solute self-polarization... 

The exp-6 model is not well suited to molecules with large dipole moments. To account for this, Ree9 used a temperature-dependent well depth e(T) in the exp-6 potential to model polar fluids and fluid phase separations. Fried and Howard have developed an effective cluster model for HF.33 The effective cluster model is valid for temperatures lower than the variable well-depth model, but it employs two more adjustable parameters than does the latter. Jones et al.34 have applied thermodynamic perturbation theory to... 

In general, all three contributions to the second order perturbation theory stabilize the molecular association AB. zJApol is the analogue to the classical polarization energy and becomes identical with the function (2 C R -n) calculated... 

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