The concept of a Potential Energy Surface (PES)

 

The PES is related to the Born-Oppenheimer approximation in molecular quantum mecanics. This approximation implies that the total molecular wavefunction is written as a product of an electronic wavefunction and a nuclear wavefunction. Let us consider a system comprising M nuclei and N electrons. By including only electrostatic interactions, the Hamiltonian of the system is given by


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in this equation r and R are used as shorthand notation for the electronic and nuclear coordinates MATH and MATH, repectively. Further, we use MATH, as a symbol for the electronic spin coordinates. All electrostatic interactions, i.e. electron-electron, electron-nuclear, nuclear-nuclear, are included in V(r,R). The mass of the nucleus $\alpha $ is denoted M$_{\alpha }$ and m$_{e}$ is the mass of the electron. The time-independent Schrödinger equation is the starting point:


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In the Born-Oppenheimer approximation the wavefunction $\Psi _{E}$ is written as a product function




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The electronic wavefunction $\psi _{e}$ is a solution of the electronic Schrödinger equation




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Since the potential depends on the nuclear coordinates, the electronic wavefunctions depends parametrically on R and the "eigenvalue" E$_{e}$ is a function of the nuclear coordinates. By replacing $\Psi _{E}$ by $\Psi ^{B.O.}$ in the Schrödinger equation for the total system, and neglecting some coupling terms, we arrive at the Schrödinger equation for the nuclear wavefunction




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where $E^{B.O.}$ denotes the energy for the system within the Born-Oppenheimer approximation. The validity of this separation between electronic and nuclear motion is due to the large ratio between electronic and nuclear masses. The last equation expresses that the nuclei move in an effective potential which is the electronic energy ( including nuclear-nuclear interaction ) as a function of the internuclear distances ( $E_{e}(\QTR{bf}{R})$ is constant with respect to a translation and/or rotation of a fixed nuclear configuration ).

The key element in Classical Molecular Dynamics is to replace a quantum mechanical description of the nuclear motion by a classical one, and where the potential energy function in the classical description is in principle the quantum mechanical potential energy surface (PES) $E_{e}(\QTR{bf}{R)}$. To determine the quantum mechanical PES for a system comprising a large number of electrons and several nuclei ( i.e. more than say 3-4 nuclei ), is out of question due to insurmountable numerical difficulties. Hence, in most applications of molecular dynamics the PES has to be constructed by semi-empirical procedures.




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