Energy, correlation

Exact anal3dic expressions for Cc(n), the correlation energy per electron of the uniform gas, are known only in extreme limits. The high-density (rs 0) limit is also the weak-coupling limit, in which 

The low-density (rg - oo) limit is also the strong coupling limit in which the uniform fluid phase is unstable against the formation of a close-packed Wigner lattice of localized electrons. Because the energies of these two phases remain nearly degenerate as rg — c , they have the same kind of dependence upon rs [56]  

The constants dg and di in (1.141) can be estimated from the Madelung electrostatic and zero-point vibrational energies of the Wigner crystal, respectively. The estimate 

The uniform electron gas is in equilibrium when the density n minimizes the total energy per electron, i.e., when 

This condition is met at = 4.1, close to the observed valence electron density of sodium. At any we have 

The four sources of error in ab initio molecular electronic calculations are (1) neglect of or incomplete treatment of electron correlation, (2) incompleteness of the basis set, (3) relativistic effects, and (4) deviations from the Born-Oppenheimer approximation. Deviations from the Born-Oppenheimer approximation are usually negligible for ground-state molecules. Relativistic effects will be discussed in Section 16.11. In calculations on molecules without heavy atoms, (1) and (2) are the main sources of error. 

Almost all computational methods expand the MOs in a basis set of one-electron functions. The basis set has a finite number of members and hence is incomplete. The incompleteness of the basis set produces the basis-set incompleteness (or truncation) error (BSIE or BSTE). 

The Hartree-Fock method, discussed in Chapter 15, neglects electron correlation. Chapter 16 discusses methods that include electron correlation. The main correlation methods are configuration interaction (Cl, Section 16.2), M0ller-Plesset (MP) pertnrbation theory (Section 16.3), the coupled cluster (CC) method (Section 16.4), and density fnnc-tional theory (DFT, Section 16.5). 

Recall from Eq. (11.16) that the molecular correlation energy is defined as the difference between the nonrelativistic true molecular energy and the restricted Hartree-Fock (HF) nonrelativistic energy calculated with a complete basis set = nonrei hf- For a 

In Section 11.6 the existence of a nonzero E orr was ascribed to the failure of the HF calculation to account for the instantaneous correlations between motions of electrons. This source of E orr is called dynamic correlation. 

A part of the electronic energy is not considered in case of ab initio SCF calculations since the electrons of different spins are treated as independent (uncorrelated) within the framework of this approach. If the corresponding energy (correlation energy) is of different magnitude in the complex and in its constituents, the correlation energy contribution to the interaction energy has to be evaluated. 

The intermolecular electron correlation (the dispersion interaction) was calculated or estimated for some cation-ligand interactions using configuration interaction (Cl) calculations, perturbation theory or on the basis of a statistical model (see Table 4). Its contribution to the total interaction energy is less than 10% throughout. 

The difference between the exact ground state energy eigenvalue of the SE, Eq, and the best (in the sense of the variation principle) energy expectation value (h) over the Slater determinant, is due to the neglect of correlation between electrons in their positions, particularly between electrons with different spins, in the Slater determinant. This was first pointed out by the Swedish quantum chemist Per-Olov Lowdin. He referred to the error of the Slater determinant as correlation energy  

In the field of ab initio calculations there exists an apparent 

The first condition is satisfied automatically with all reactions containing closed shell molecules only. A systematic examination for this type of reactions was performed by Snyder and Basch The theoretical (SCF) heats of reactions were claimed to be more accurate than those obtained using seraiempirical relations of bond energies for reactions of strained molecules, or those not well represented by a single valence-bond structure. However, Snyder and Basch concluded 

Homodesmotic reactions. These are reactions in which (i) there are equal numbers of bonds of a particular type (e.g., C[4]-C[4], C[4]-C[3], C[3]-C[3], C[3j C[3], where the numbers in brackets indicate the total numbers of other atoms bonded to each carbon atom, so that C[4]-C[4] and C[3]=C[3] may represent CC bonds in ethane and ethylene, respectively) and (ii) there are equal numbers of each atomic type (such as C[4], C[3j, etc.) with zero, one, two and three hydrogen atoms attached in reactants and products. The following exam-pies may be given  

In homodesmotic reactions the structural elements in reactants and products match closely so that only a small change in correlation energy is to be expected. Indeed, for processes of the type (4.1) and 

Isodesmic reactions. This concept introduced by Hehre and colla-borators means the reactions in which there is retention of the number of bonds of a given formal type, but with a change in their re-la tion to one another. The homodesmotic processes are actually a subclass of isodesmic reactions. Typical representants of isodesmic reactions are so called bond separation reactions such as for example 

Sometimes the term restricted Hartree-Fock (RHF) is used to emphasize that the wavefunction is restricted to be a single determinantal function for a configuration wherein electrons of a spin occupy the same space orbitals as do the electrons of P spin. When this restriction is relaxed, and different orbitals are allowed for electrons with different spins, we have an unrestricted Hartree-Fock (UHF) calculation. This refinement is most likely to be important when the numbers of a- and -spin electrons differ. We encountered this concept in Section 8-13, where we noted that the unpaired electron in a radical causes spin polarization of other electrons, possibly leading to negative spin density. 

The Hartree-Fock energy is not as low as the tme energy of the system. The mathematical reason for this is that our requirement that be a single determinant is restrictive and we can introduce additional mathematical flexibility by allowing i/r to contain many determinants. Such additional flexibility leads to further energy lowering. 

There is a corresponding physical reason for the HF energy being too high. It is connected with the independence of the electrons in a single determinantal wavefunction. To understand this, consider the four-electron wavefunction 

Because electrons repel each other, there is a tendency for them to keep out of each other s way. That is, in reality, their motions are correlated. The HE energy is higher than the true energy because the HE wavefunction is formally incapable of describing correlated motion. The energy difference between the HE and the exact (for a simplified nonrelativistic hamiltonian) energy for a system is referred to as the correlation energy. 

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Although it is now somewhat dated, this book provides one of the best treatments of the Hartree-Fock approximation and the basic ideas involved in evaluating the correlation energy. An especially valuable feature of this book is that much attention is given to how these methods are actually implemented. 

In contemporary theories, a is taken to be and correlation energies are explicitly included in the energy... 

Table 3.1.1 Pair correlation energies for the four electrons in Be.

The orbitals from which electrons are removed can be restricted to focus attention on the correlations among certain orbitals. For example, if the excitations from the core electrons are excluded, one computes the total energy that contains no core correlation energy. The number of CSFs included in the Cl calculation can be far in excess of the number considered in typical MCSCF calculations. Cl wavefimctions including 5000 to 50 000 CSFs are routine, and fimctions with one to several billion CSFs are within the realm of practicality [53]. 

Krishnan R and Pople J A 1978 Approximate fourth-order perturbation theory of the electron correlation energy Int. J. Quantum Chem. 14 91-100... 

Professor Axel Becke of Queens University, Belfast has been very actively involved in developing and improving exchange-correlation energy functionals. For a good recent overview, see ... 

Specifies the calculation ofelectron correlation energy using the Mwllcr-i lessct second order perturbation theory (Ml 2). This option can only be applied Lo Single Point calculations. 

MP2 correlation energy calculations may increase the computational lime because a tw o-electron integral Iran sfonnalion from atomic orbitals (.40 s) to molecular orbitals (MO s) is ret]uired. HyperClicrn rnayalso need additional main memory arul/orcxtra disk space to store the two-eleetron integrals of the MO s. 

HyperChem supports MP2 (second order Mdllcr-l Icsset) correlation energy calcu latiou s u sin g any available basis set. lu order to save main memory and disk space, the HyperChem MP2 electron correlation calculation normally uses a so called frozen-core approximation, i.e. th e in n er sh el I (core) orbitals are omitted. A sett in g in CHHM.IX I allows excitation s from th e core orbitals to be include if necessary (melted core). Only the single poin t calcula-tion is available for this option. 

HyperChein perforins ab initio. SCK calculations generally. It also can calculate the coi relation energy (to he added to the total -SCK energy) hy a post Hartree-Fock procedure call. M P2 that does a Moller-Plesset secon d-order perturbation calculation. I he Ml 2 procedure is on ly available for sin gle poin t calculation s an d on ly produces a single tiuin ber, th e Ml 2 correlation energy, to be added to the total SCF en ergy at th at sin gle poin t con figuration of th e ti iiclei. 

In Ecjuation (3.47) we have written the external potential in the form appropriate to the interaction with M nuclei. , are the orbital energies and Vxc is known as the exchange-correlation functional, related to the exchange-correlation energy by ... 

This result applies when the number of up spins equals the number of down spins and so is not applicable to systems with an odd number of electrons. The correlation energy functional was also considered by Vosko, Wdk and Nusarr [Vosko et al. 1980], whose expression is ... 

L. iitortunately, this simple approach does not work well, but Becke has proposed a strategy which does seem to have much promise [Becke 1993a, b]. In his approach the exchange-correlation energy Exc is written in the following form ... 

Lee C, W Yang and R G Parr 1988. Development of the Colle-Salvetti Correlation Energy Formula into a Functional of the Electron Density. Physical Review B37 785-789. 

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