![]() Such calculations can therefore be expensive. One point of warning should be noticed here: electron correlation treatments require much larger one-electron basis sets than Hartree-Fock or DFT to yield converged results. Also density functional (DFT) methods take into account electron correlation, even though in a less systematic and less well defined way than ab initio methods. ![]() There are many different methods available and implemented in Molpro to approximate and optimize the wavefunction, for instance Møller-Plesset (MP) perturbation theory, configuration interaction (CI), or coupled cluster (CC) methods. The corresponding energy lowering is called electron correlation energy. The purpose of post-Hartree-Fock electron correlation methods is to correct for this by taking the instantaneous correlation of the electrons into account. Thus, even though the Coulomb interaction between the electrons is taken into account in an averaged way, the electrons with opposite spin are unable to avoid each other when they come close, and therefore the electron repulsion is overestimated in Hartree-Fock. In the Hartree-Fock approximation each electron moves in an average potential of the remaining electrons, but has no knowledge of the positions of these. This is the Hartree-Fock (HF) self consistent field (SCF) method, and it is usually the first step in any ab initio calculation. The simplest choice is to use a single Slater determinant and to optimize the orbitals variationally. Once such approximations are introduced, it matters how the orbitals are determined. Therefore, approximations have to be made, in which the wavefunction is expanded in only a subset of all possible of Slater determinants (or configuration state functions (CSFs), which are symmetry adapted linear combinations of Slater determinants). ![]() However, the number of Slater determinants which can be constructed is enormous, and very quickly increases with the number of electrons and orbitals. In a full configuration interaction calculation (FCI) all possible Slater determinants for a given orbital basis are used, and this gives the best possible result for the chosen one-electron basis. The many-electron wavefunction for the molecule is represented as a linear combination of antisymmetrized products (Slater determinants) of the molecular orbitals. ![]() Many optimized basis sets are available in the Molpro basis set library, and in most cases the basis set can be selected using a simple keyword in the input. In most programs, and also in Molpro, Gaussian basis functions are used to approximate the molecular orbitals, since the required integrals can be computed very quickly in this basis. There are two classes of approximations: one concerning the choice of basis functions to represent the one-electron functions called molecular orbitals, and one concerning the choice of $N$–electron functions to represent the many-electron electronic wavefunction. Therefore, the electronic wavefunction is represented in certain finite basis sets, and the Schrödinger equation is transformed into an algebraic equation which can be solved using numerical methods. Secondly, the electronic Schrödinger equation cannot be solved exactly, except for very simple systems like the hydrogen atom. The aim of most calculations is to find these structures and to characterize the potential and the molecular properties in the vicinity of the stationary points of the PES. The minima correspond to equilibrium structures of different isomers or molecules, and saddle points to transition states between them. The PES is in general very complicated and can have many minima and saddle points. The electronic energy as function of the 3N-6 internal nuclear degrees of freedom defines the potential energy surface (PES). The ab initio program like Molpro then computes the electronic energy by solving the electronic Schrödinger equation for this fixed nuclear configuration. Thus, each electronic structure calculation is performed for a fixed nuclear configuration, and therefore the positions of all atoms must be specified in an input file. We just want to remind you of some basic approximations, which are made in any ab initio calculation, independent of which program is used.įirstly, the Born-Oppenheimer approximation is applied, which means that the nuclear and electronic motions are decoupled and treated separately (in some cases, non-adiabatic couplings are taken into account at a later stage). Of course, this manual cannot teach you the underlying theory, and it is assumed that you are familiar with it. The electronic structure of molecules can be treated only by quantum mechanics, since the electrons are very quickly moving particles.
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