Clamped nuclei 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. 

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. 

In eq.(44) we have indicated the parametric dependences which are directly relevant to the discussion. In the clamped nuclei approximation, 4>AT,n( G R-)... 

To start with, consider systems consisting of N dynamical electrons and positrons and K fixed nuclei with Coulomb interactions between all pairs of particles. The clamped-nuclei approximation (the Bom-Oppenheimer approximation) may be legitimate because of the huge difference in mass between electrons and nuclei. Stability of matter means that the energy of such a model system is bounded from below by a negative constant times the number of particles E —C(N 4- K). Such a condition is necessary for some basic physical properties such as the existence of the thermodynamical limit. 

With these approximations, the electronic structure can be treated using a relativistic quantum mechanical description, while the nuclei are held fixed. To overcome this clamped-nuclei approximation, an attempt has been made (Parpia et al. 1992b) to include relativistic corrected nuclear motion terms and to reach an adiabatic separation of the electronic and nuclear motion at least for atoms. 

As mentioned in the introduction, the calculations were initially carried out by using the well-known clamped nuclei approximation for the nuclear dynamics which is the basis for the Bom-Oppenheimer (BO) decoupling approach. The actual basis set chosen was the one corresponding to a triple-zeta expansion plus polarisation usually labelled by the acronym cc-pVTZ in the current literature [29]. Thus, our clamped nuclei calculations will be labelled throughout the present work as B3LYP/cc-pVTZ calculations. 

But this trouble disappears after the Bom-Oppenheimer approximation (the clamped nuclei approximation, cf. p. 268) is used i.e., if we hold the molecule fixed in space. In such a case, the molecule has the dipole moment, and this dipole moment is to be inserted into formulas as /tQ, and then we may calculate tiie polarizability, hyperpolarizabilities, etc. (see p. 72). But what do we do when we do not apply the Born-Oppenheimer approximation Yet, in experiments, we do not use the Born-Oppenheimer approximation (or any other one). We have to allow the molecule to rotate and then the dipole moment fiQ disappears. Well, that is not quite true since the space is no longer isotropic. There is a chance to measure a dipole moment What do we measure, then ... 

Adjustable parameters in the approximate wavefunction F can be varied to minimize Evar. yielding an improved energy and wavefunction. The Bom-Oppenheimer approximation (clamped nuclei approximation) is used to separate the electronic and the nuclear motions. Further restrictions on the form of the electronic wavefunction and the manner in which the adjustable parameters are determined are governed by level of theory used. 

In many applications the positions of the nuclei are fixed (clamped nuclei approximation, Chapter 6), often in a high-symmetry configuration (cf. Appendix C, p. 903). For example, the benzene molecule in its ground state (after minimizing the energy with respect to the positions of the nuclei) has the symmetry of a regular hexagon. In such cases the electronic Hamiltonian additionally exhibits invariance with respect to some symmetry operations and therefore the wave functions are... 

In order to simplify the Hamiltonian of Eq. (8.66), we freeze the nuclear motion and can then neglect the kinetic energy operators for the nuclei. In this clamped-nuclei approximation the remaining electronic Hamiltonian reads. 

The two-electron interaction operators g i,j) in the many-electron Hamiltonian are the reason why a product ansatz for the electronic wave function Yg/( r, ) that separates the coordinates of the electrons is not the proper choice if the exact function is to be obtained in a single product of one-electron functions. Instead, the electronic coordinates are coupled and the motion of electrons is correlated. It is therefore natural to assume that a suitable ansatz for the many-electron wave function requires functions that depend on two coordinates. Work along these lines within the clamped-nuclei approximation has a long history [320-328]. The problem, however, is then what functional form to choose for these functions. First attempts in molecular quantum mechanics used simply terms that are linear in the interelectronic distance Tij = ki — Tjl [329-333], while it could be shown that an exponential ansatz is more efficient [334-337]. 

In the following we shall be concerned with the problem of the electronic structure of molecular systems containing N electrons and M nuclei. The validity of the usual Born-Oppenheimer, or clamped nuclei, approximation will be assumed, that is we shall investigate the distribution of electrons in the field of the fixed nuclei. In principle the approximate solution of the Schrddinger equation of all the electrons provides us with the different electronic states of the molecule, once the position of the nuclei and the number of electrons is given. Essentially this is the procedure followed in everyday routine calculations of ab initio quantum chemistry, where we do not take into account the a priori knowledge about the properties of the different fragments of the total composite system. 

The atomic SCF calculations described in Section 2.3 may be extended in principle to diatomic molecules with closed-shell electron configurations. The diatomic electronic Hamiltonian in the clamped-nuclei approximation (Eqs. 3.7 and 3.9) may be broken down into a sum of one-electron operators and electron repulsion terms... 

There is clearly a choice between the form ( Eqs. 2.24 and O 2.34) and although in the clamped nuclei approximation both would yield the same energies for any chosen internudear separation a, the resulting energy would be a potential for two quite different situations. 

This is not for any sinister reason but simply because of the definition chosen of the clamped nuclei approximation. In this approximation the nuclear positions are triplets of numbers which take particular but fixed values once a particular basis is chosen. ... 

In equation (1) Eq R) is the energy obtained by solving the electronic Schrodinger equation in the clamped nuclei approximation ... 

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