Clamped-nucleus

Here, t is the nuclear kinetic energy operator, and so all terms describing the electronic kinetic energy, electron-electron and electron-nuclear interactions, as well as the nuclear-nuclear interaction potential function, are collected together. This sum of terms is often called the clamped nuclei Hamiltonian as it describes the electrons moving around the nuclei at a particular configrrration R. 

Ah initio calculation s can be performetl at th e Ilartree-Fock level of approximation, equivalent to a self-con sisten t-field (SCK) calculation. or at a post llartree-Fock level which includes the effects of correlation —defined to be everything that the Hartree-Fock level of appi oxiniation leaves out of a n on-relativistic solution to the Schrddinger ec nation (within the clamped-nuclei Born-Oppenh e-imer approximation ). 

The first basic approximation of quantum chemistry is the Born-Oppenheimer Approximation (also referred to as the clamped-nuclei approximation). The Born-Oppenheimer Approximation is used to define and calculate potential energy surfaces. It uses the heavier mass of nuclei compared with electrons to separate the... 

Molecules in their ground state are typically treated using the so-called Born-Oppen-heimer approximation. This approximation is also known as the clamped nuclei approximation because it views the electrons as moving in a field of fixed nuclei. In other words, the total wave function, which is a function of nuclear and electronic coordinates, can be separated into a nuclear wave function and an electronic wave function. This approximation can be justified on the basis that electrons move much faster than nuclei and follow them quasi-instantly. 

Technically, the time-independent Schrodinger equation (2) is solved for clamped nuclei. The Hamiltonian is broken into its electronic part, He, including the nuclear Coulomb repulsion energy, and the nuclear Hamiltonian HN. At this level, mass polarization effects are usually neglected. The wave function is therefore factorized as usual (r,X)= vP(r X)g(X). Formally, the electronic wave function d lnX) and total electronic energy, E(X), are obtained after solving the equation for each value of X ... 

With the triples correction added, the error relative to experiment is still as large as 15 kJ/mol. More importantly, we are now above experiment and it is reasonable to assume that the inclusion of higher-order excitations (in particular quadruples) would increase this discrepancy even further, perhaps by a few kJ/mol (judging from the differences between the doubles and triples corrections). Extending the coupled-cluster expansion to infinite order, we would eventually reach the exact solution to the nonrelativistic clamped-nuclei electronic Schrodinger equation, with an error of a little more than 15 kJ/mol. Clearly, for agreement with experiment, we must also take into account the effects of nuclear motion and relativity. 

Treating the electronic Schrodinger equation in the usual clamped nuclei (Bom-Oppenheimer) approximation, [20] we have (in atomic units) the Hamiltonian, H, and the spectrum of eigenvalues and eigenvectors, and fife,... 

We can make further approximations to simplify the NRF of the Hamiltonian presented in equation (75) for non-dynamical properties. For such properties, we can freeze the nuclear movements and study only the electronic problem. This is commonly known as the clamped nuclei approximation, and it usually is quite good because of the fact that the nuclei of a molecule are about 1836 times more massive than the electrons, so we can usually think of the nuclei moving slowly in the average field of the electrons, which are able to adapt almost instantaneously to the nuclear motion. Invocation of the clamped nuclei approximation to equation (75) causes all the nuclear contributions which involve the nuclear momentum operator to vanish and the others to become constants (nuclear repulsion, etc.). These constant terms will only shift the total energy of the system. The remaining terms in the Hamiltonian are electronic terms and nuclear-electronic interaction contributions which do not involve the nuclear momentum operator. 

The distinction between interaction of different zero order electronic states for clamped nuclei, and interaction of different Bom-Oppenheimer states due to the... 

The terms, Enn(Q)and Hpq(Q) are introduced over the Taylor s expansion at the equilibrium geometry of fixed nuclear configuration Ro (crude level -clamped nuclei),... 

In this equation, HA(crude) is the clamped nuclei (crude - BO level) Hamiltonian,... 

The clamped-nuclei model is only valid under the assumption that the two protons behave classically and the electron quantum-mechanically. The rationale to justify this assumption is based on the difference in mass between proton and electron. For all practical purposes the protons should therefore... 

We shall show how the Bom-Oppenheimer potential energy for the Hj ion can be calculated exactly using series expansion methods, even though an exact analytical solution cannot be obtained. Figure 6.29 shows the coordinate system used for an electron moving in the field of two clamped nuclei. In atomic units the Hamiltonian is... 

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