Hartree LCAO approximation

The LCAO approximation for the wave functions in the Hartree-Fock equations... 

Configuration Interaction (or electron correlation) adds to the single determinant of the Hartree-Fock wave function a linear combination of determinants that play the role of atomic orbitals. This is similar to constructing a molecular orbital as a linear combination of atomic orbitals. Like the LCAO approximation. Cl calculations determine the weighting of each determinant to produce the lowest energy ground state (see SCFTechnique on page 43). 

Problem 11-1. Consider three levels of approximation (a) Exact many-electron wave function, (b) Hartree-Fock wave function, (including all electrons), (c) Simple LCAO-MO valence electron wave function. For each of the following molecular properties, would you expect the Hartree-Fock approximation to give a correct prediction (to within 1% in the cases of quantitative predictions) Would you expect the LCAO-MO approximation to give a correct prediction ... 

The Hartree-Fock and LCAO approximations, taken together and applied to the electronic Schrodinger equation, lead to the Roothaan-Hall equations. ... 

Here, /i are the so-called Kohn-Sham orbitals and the summation is carried out over pairs of electrons. Within a finite basis set (analogous to the LCAO approximation for Hartree-Fock models), the energy components may be written as follows. 

LCAO Approximation. Linear Combination of Atomic Orbitals approximation. Approximates the unknown Hartree-Fock Wavefunctions (Molecular Orbitals) by linear combinations of atom-centered functions (Atomic Orbitals) and leads to the Roothaan-Hall Equations. 

Roothaan-Hall Equations. The set of equations describing the best Hartree-Fock or Single-Determinant Wavefunction within the LCAO Approximation. 

The Hartree-Fock equations are the coupled differential equations of the SCF procedure. The LCAO approximation transforms these differential equations into an ensemble of algebraic equations, which are substantially easier to solve. 

The idea to employ a finite basis set of AOs to represent the MOs as linear combinations of the former apparently belongs to Lennard-Jones [68] and had been employed by Hiickel [37] and had been systematically explored by Roothaan [38]. That is why the combination of the Hartree-Fock approximation with the LCAO representation of MOs is called the Hartree-Fock-Roothaan method. 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

As we have noted, electronic stracture techniques attempt to solve the Schrodinger equation. The traditional approach in quantum chemistry has been to use the Hartree Fock (HF) approximation, in which a determinantal, antisymmetrized wave function is optimized in accordance with the variational principle. The wave function is normally written as an expansion of atomic orbitals (the LCAO approximation). A major weakness of the HF method is that in its single... 

The first ab initio calculations on interaction dipoles were performed by Matcha and Nesbet They considered the systems HeNe, HeAr and NeAr as super-molecules and did ordinary Hartree-Fock-LCAO-SCF calculations in the range R = 2.0 to 5.5aQ. Because of the Hartree-Fock approximation they did not obtain the dispersion contribution to the dipole moment (cf. sect. 2), but only exchange, penetration and overlap-induction contributions. Their ab initio dipoles could be fitted quite well by a single exponential, which supported the assumption made earlier by Van Kranendonk ... 

Both electrons occupy this bonding orbital, satisfying the condition of indistin-guishability and the Pauli principle. Recall from Section 6.2 that the electronic wave function for the entire molecule in the LCAO approximation is the product of all of the occupied MOs, just as an atomic wave function is the product of all occupied Hartree orbitals of an atom. Thus, we get... 

The extension of the basis can improve wave functions and energies up to the Hartree-Fock limit, that is, a sufficiently extended basis can circumvent the LCAO approximation and lead to the best molecular orbitals for ground states. However, this is still in the realm of the independent-particle approximation 175>, and the use of single Slater-determinant wave functions in the study of potential surfaces implies the assumption that correlation energy remains approximately constant on that part of the surface where reaction pathways develop. In cases when this assumption cannot be accepted, extensive configuration interaction (Cl) must be included. A detailed comparison of SCF and Cl results is available for the potential energy surface for the reaction F + H2-FH+H 47 ). 

To discuss the form and cost of analytic gradient and Hessian evaluations, we consider the simple case of Hartree-Fock (HF) calculations. In nearly all chemical applications of HF theory, the molecular orbitals (MOs) are represented by a linear combination of atomic orbitals (LCAO). In the context of most electronic structure methods, the LCAO approximation employs a more convenient set of basis functions such as contracted Gaussians, rather than using actual atomic orbitals. Taken together, the collection of basis functions used to represent the atomic orbitals comprises a basis set. 

Generally speaking, it is probably impossible to give any general recommendations concerning the search for the observables, the operators of which commute with the Hamiltonian. The exceptions are the cases when the observable to be found characterizes the properties of the spatial symmetry of the ket-vector v /(t)> (in Schroedinger s spatial presentation this vector is called the wave function). It should be noted that a more or less precise pattern of the wave function /(t) is known for very few molecular system. At present the most widespread is the i /(t) presentation in the Hartree-Fock approximation as a symmetrized linear combination of atomic orbitals (LCAO) [16]. 

The virtual orbitals generated by the solution of the LCAO approximation to the Hartree-Fock equations are indeed an artifact of the LCAO technique and do not have any physical interpretation except as a residue of those features of the basis functions which are not suitable for the description of the single-determinant model of the electronic structure of the molecule. 

This method should lead to results which are just as accurate as the results of the methods described in the previous sections, and can be used as a check on the computed potential-energy minimum E(R ) at R = Re if fl is determined from curve-fitting of the Morse potential with the computed R and De and this leads to a wrong we and/or w, then it can be assumed that De and/or Rg are/is wrong. It is to be emphasized (12) that the Morse curve can mostly not be used with essentially ionic compounds like NaF because the attraction given by the Coulomb term extends out in space to greater distances than the Morse exponential part for these compounds many other types of potential have been postulated (e.g. the Hellmann-potential or the Bom-Landd potential (77)). The reader can try to calculate cog, etc. of NaF from the SCF— LCAO—MO calculation of Matcha (72) in the Roothaan-Hartree-Fock approximation, using the Morse curve (E = —261.38 au, R =3.628 au experimental values Rg = 3.639 au, a)g=536 cm i, >g g=3.83 cm-i). 

The LCAO approximation was introduced to the Hartree-Fock method, independently, by C.CJ. Roothaan and G.G. Hall. 

The many-electron theory (e.g., Hartree-Fock-Roothan) of molecular systems within the LCAO approximation involves the electron repulsion integrals of the type which represent... 

In Sect. 4.1.5 the Hartree-Fock LCAO approximation for periodic qrstems was considered. The main difference of the CO LCAO method (crystalline orbitals as linear combination of atomic orbitals) from that used in molecular quantum chemistry, the MO LCAO (molecular orbitals as hnear combination of atomic orbitals) method was explained. In the CO LCAO approximation the one-electron wavefunction of a crystal (CO - ifih R)) is expanded in Bloch sums Xt kiR) of AOs ... 

In the linear combination of atomic orbitals (LCAO) approximation of the molecular orbital, the energy of the overlap electron density between the atomic orbitals Xu and Xv due to attraction by the core. See Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Relativistic Effective Core Potential Techniques for Molecules Containing Very Heavy Atoms Transition Metal Chemistry and Transition Metals Applications. 

Fig. 2.1 The matrix elements Sab = S,Hab,Haa, the reduced resonance integral and energy eigenvalues of as functions of nuclear distance. The equilibrium value corresponds to the minimum of in the LCAO approximation used (ao = Bohr unit of length = 0.529 A Eq = Hartree energy unit = 27.21 eV). FVom Ref. [20].

The fundamental tool for the generation of an approximately transferable fuzzy electron density fragment is the additive fragment density matrix, denoted by Pf for an AFDF of serial index k. Within the framework of the usual SCF LCAO ab initio Hartree-Fock-Roothaan-Hall approach, this matrix P can be derived from a complete molecular density matrix P as follows. 

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