Perturbation theory potential energy function

Free energy calculations rely on a well-known thermodynamic perturbation theory [6, 21, 22], which is recalled in Chap. 2. We consider a molecular system, described by the potential energy function U(rN), which depends on the coordinates of the N atoms rN = (n, r2,..., r/v). The system could be a biomolecule in solution, for example. We limit ourselves to a classical mechanical description, for simplicity. Practical calculations always consider differences between two or more similar systems, such as a protein complexed with two different ligands. Therefore, we consider a change in the system, such that the potential energy function becomes ... 

As in the BK procedure, the electrical properties and the potential energy surface may be expanded as a Taylor series in the normal coordinates. Orders of perturbation theory are defined in the same way as for the non-resonant case. Electrical property terms that are quadratic, cubic,. .. in the normal coordinates are taken to be first-order, second-order,. .. terms in the potential energy function that are... 

The central idea of thermodynamic perturbation theory is that the potential energy function can be partitioned in a convenient way i.e., one can write... 

In many multireference perturbation theories, the energy spectrum computed with the approximated Hamiltonian is ill conditioned. Functions outside the reference space become artificially degenerate with reference wavefunction. The phenomenon is known in the literature as intruder state, and it results in unphysical energy corrections and spurious bumps along a potential energy surface. Several schemes, such as ad hoc level shift parameter, have been proposed [97-100], yet... 

In using Eqs. 19 and 34 as starting points for the development of the theory of transport processes, several tasks remain. The first and perhaps simplest is to develop expressions for the transport coefficients in terms of the single and pair densities /(1> and /< >. Next the continuity equations must be solved to give explicit expressions for the densities when the system is perturbed from its equilibrium state by the transport process under consideration. Finally, it is required to obtain the frictional coefficient f in terms of the intermolecular potential energy function according to Eq. 33. 

At this point the first-principles perturbative (FP) approach becomes valuable. The same kinds of perturbative models are used to describe the vibrational-rotational motions as in the SP approach. However, data from electronic structure theory computations or potential energy functions are used to parameterize the formulas instead of spectroscopically obtained data. The FP approach has for example, been pursued by Martin et al. [16-18] and by Isaacson, Truhlar, and co-workers [19-25]. This avenue is especially valuable when spectroscopic data are not available for a molecule of interest. Codes are available that can carry out vibrational perturbation theory computations, using a grid of ab initio data as input SURVIBTM... 

In practice, however, these calculations are rather intractible for many-electron molecules, so that, using the Bom-Oppenheimer approximation (see below), it is assumed that the electronic motion can be factored off from Eq. (1.1), yielding an effective potential-energy field in which the nuclear motions (vibrations) take place and which couple with the rotatory motion of the molecule. Normally perturbation theory is used in the first and second approximations to give the vibrational and rotational eigenvalues of H. Instead of a theoretically-obtained potential-energy function (see below), it is normally assumed that the vibratory motion follows Hooke s law (harmonic oscillator) with... 

A many-body perturbation theory (MBPT) approach has been combined with the polarizable continuum model (PCM) of the electrostatic solvation. The first approximation called by authors the perturbation theory at energy level (PTE) consists of the solution of the PCM problem at the Hartree-Fock level to find the solvent reaction potential and the wavefunction for the calculation of the MBPT correction to the energy. In the second approximation, called the perturbation theory at the density matrix level only (PTD), the calculation of the reaction potential and electrostatic free energy is based on the MBPT corrected wavefunction for the isolated molecule. At the next approximation (perturbation theory at the energy and density matrix level, PTED), both the energy and the wave function are solvent reaction field and MBPT corrected. The self-consistent reaction field model has been also applied within the complete active space self-consistent field (CAS SCF) theory and the eomplete aetive space second-order perturbation theory. ... 

FIGURE 4.6 Perturbation theory used to find an approximate formula for the shape of water at the bottom of a pond, (a) The pits and mounds in the earth at the pond bottom establish a potential energy function, in which points at lower height have lower potential energy, (b) We break the function up into two pieces a zero-order piece that has the right general shape and obeys a simple formula—for example, the formula for a parabola—and a perturbation, which contains all the complicated parts. For the zero-order part, we can solve the problem exactly (the distribution of the water in this limit is known, but not very accurate). 

If the energy and angular momentum of the diatomic are not too large and the potential energy function for the diatomic is either the harmonic or Morse function, n can be determined from the following second-order perturbation theory expression for Ed( ,7) ... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Curve 1 represents the total energy of the hydrogen molecule-ion as calculated by the first-order perturbation theory curve 2, the naive potential function obtained on neglecting the resonance phenomenon curve 3, the potential function for the antisymmetric eigenfunction, leading to elastic collision. 

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