2 How an ab initio calculation works

The electronic structure of molecules can be treated only by quantum mechanics, since the electrons are very quickly moving particles. Of course, this manual cannot teach you the underlying theory, and it is assumed that you are familiar with it. We just want to remind you of some basic approximations, which are made in any ab initio calculation, independent of which program is used.

Firstly, 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). 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. The ab initio program like MOLPRO then computes the electronic energy by solving the electronic Schrödinger equation for this fixed nuclear configuration. The electronic energy as function of the 3N-6 internal nuclear degrees of freedom defines the potential energy surface (PES). The PES is in general very complicated and can have many minima and saddle points. The minima correspond to equilibrium structures of different isomers or molecules, and saddle points to transition states between them. 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.

Secondly, the electronic Schrödinger equation cannot be solved exactly, except for very simple systems like the hydrogen atom. 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. 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.

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. 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.

The many-electron wavefunction for the molecule is represented as a linear combination of antisymmetrized products (Slater determinants) of the molecular 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. However, the number of Slater determinants which can be constructed is enormous, and very quickly increases with the number of electrons and orbitals. 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).

Once such approximations are introduced, it matters how the orbitals are determined. The simplest choice is to use a single Slater determinant and to optimize the orbitals variationally. This is the Hartree-Fock (HF) self consistent field (SCF) method, and it is usually the first step in any ab initio calculation.

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. 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. The purpose of post-Hartree-Fock electron correlation methods is to correct for this by taking the instantaneous correlation of the electrons into account. The corresponding energy lowering is called electron correlation energy. 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. 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.

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. Such calculations can therefore be expensive. For a fixed basis set, a correlation calculation is usually much more expensive than a HF calculation, and therefore many unexperienced people are tempted to use small basis sets for a correlation calculation. However, this is not reasonable at all, and for meaningful calculations one should at least use a triple-zeta basis with several polarization functions (e.g. cc-pVTZ). Using explicitly-correlated methods discussed later, the basis set problem is much alleviated.

Finally, it should also be noted that the HF approximation, and all single reference methods which use the HF determinant as zeroth order approximation, are usually only appropriate near the equilibrium structures. In most cases they are not able to dissociate molecular bonds correctly, or to describe electronically excited or (nearly) degenerate states. In such cases multireference methods, which use a multiconfiguration SCF wavefunctions (MCSCF) as zeroth order approximation, offer a reasonable alternative. Complete active space SCF (CASSCF) is a special variant of MCSCF. In MOLPRO various multireference electron correlation methods are implemented, e.g., multireference perturbation theory (MRPT, CASPT2) and multireference configuration interaction (MRCI), and variants of these such as multireference coupled-pair functional (MR-ACPF).

As you will see, it is quite easy to run an electronic structure calculation using MOLPRO, and probably you will have done your first successful run within the next 10 minutes. However, the art is to know which basis set and method to use for a particular problem in order to obtain an accurate result for a minimum possible cost. This is something which needs a lot of experience and which we cannot teach you here. We can only encourage you not to use MOLPRO or any other popular electronic structure program simply as a black box without any understanding or critical assessment of the methods and results!

Literature: Introduction to Computational Chemistry, F. Jensen, Wiley, 2006

molpro@molpro.net 2019-03-22