Hartree electron correlation energy

We can compute all of the results except those in the first row by running just three jobs QCISD(T,E4T] calculations on HF and fluorine and a Hartree-Fock calculation on hydrogen (with only one electron, the electron correlation energy is zero). Note that the E4T option to the QCISDfT) keyword requests that the triples computation be included in the component MP4 calculation as well as in the QCISD calculation (they are not needed or computed by default). 

With sufficiently large basis set, the Hartree-Fock (HF) method is able to account for 99% of the total energy of the chemical systems. However, the remaining 1% is often very important for describing chemical reaction. The electron correlation energy is responsible for the same. It is defined as the difference between the exact nonrelativistic energy of the system ( 0) and Hartree-Fock energy (E0) obtained in the limit that the basis set approaches completeness [36] ... 

Electron correlation is inherently a multi-electron phenomenon, and we believe that the retention of explicit two-electron information in the Wigner intracule lends itself to its description (i). It has been well established that electron correlation is related to the inter-electronic distance, but it has also been suggested (4) that the relative momentum of two electrons should be considered which led us to suggest that the Hartree-Fock (HF) Wigner intracule contains information which can yield the electron correlation energy. The calculation of this correlation energy, like HF, formally scales as N. ... 

The term "electron correlation energy" is usually defined as the difference between the exact nonrelativistic energy and the energy provided by the simplest MO wave function, the mono-determinantal Hartree-Fock wave function. This latter model is based on the "independent particle" approximation, according to which each electron moves in an average potential provided by the other electrons [14]. Within this definition, it is customary to distinguish between non dynamical and dynamical electron correlation. 

Well-known procedures for the calculation of electron correlation energy involve using virtual Hartree-Fock orbitals to construct corresponding wavefunctions, since such methods computationally have a good convergence in many-body perturbation theory (MBPT). Although we know the virtual orbitals are not optimized in the SCF procedure. Alternatively, it is possible to transform the virtual orbitals to a number of functions. There are some techniques to do such transformation to natural orbitals, Brueckner orbitals and also the Davidson method. 

Whatever physical reasons may exist for the correlated behaviour of the electrons—electron repulsion or Pauli anti symmetry principle—the effect is always to modify the electron-repulsion energy calculated from the electron distribution of the system. In the Hartree-Fock (HF) approximation one solves equations describing the behaviour of each electron in the averaged filed of the remaining (n—1) electrons. However the motions of the [n(n—1)]/2 pairs of electrons are correlated and the electron correlation energy is defined as... 

Finite basis set Hartree-Fock calculations yield not only an approximation for the occupied orbitals but also a representation of the spectrum which can be used in the treatment of correlation effects. In particular, the use of finite basis sets facilitates the effective evaluation of the sum-over-states which arise in the many-body perturbation theory of electron correlation effects in atoms and molecules. Basis sets have been developed for low order many-body perturbation theoretic treatments of the correlation problem which yield electron correlation energy components approaching the suh-milliHartree level of accuracy [20,21,22]. 

Basis effects, on the other hand, are known to play an importsmt role. In order to illustrate their importance we show in Fig. 1 a large change in the basis error on going from E to E. The only way to avoid this kind of situation is to choose large and flexible basis sets in order to come sufficiently close to the Hartree-Fock limit. Otherwise there is a tendency to attribute to electron correlation energy effects which have nothing to do with it. 

As discussed below, in the Hartree-Fock model each electron sees only the average field of the other electrons. In reality, the electrons must explicitly avoid each other because of their mutual coulombic repulsion hence their motions are correlated. The difference between the Hartree-Fock energy and the exact, non-relativistic energy is termed electron correlation energy. The Hartree-Fock wavefunction can be improved by taking a linear combination of Slater determinants, yielding a configuration interaction (Cl) wavefunction ... 

In most cases, the Hartree-Fock method provides a qualitatively correct description of the electronic structure of a molecular system. Usually, the Hartree-Fock method gives 99 % of the total energy of the molecule described by the non-relativistic Schrddinger equation and the clamped nucleus Hamiltonian. The difference between the best Hartree-Fock energy, i.e the Hartree-Fock energy in the limit of an infinite basis, and the exact energy is called the electronic correlation energy. 

---