System Hartree atomic

When we consider the predicted atomization energy, however, we see vast differences among the functionals. Like Hartree-Fock theory, the SVWN and SVWN5 functionals are completely inadequate for predicting this system s atomization energy (which is not an atypical result). The BLYP value is also quite poor. 

In the physics and chemistry of many-particle systems comprising atoms, molecules, solid state, and nuclei, quantum mechanics has given very important contributions to the theory of both a qualitative and quantitative nature. The Hartree-Fock scheme has usually been considered as a rather sophisticated approach, but, if one seriously studies the typical errors listed in Tables I and II and Eqs. 11.83 and 11.84 it becomes clear that the qualitative aspects... 

We consider systems consisting of two electrons and an infinitely massive nucleus of charge Z, moving subject to the nonrelativistic Hamiltonian (in hartree atomic units, me = it = c = 47r o= 1)... 

Unless otherwise stated, the atomic system of units is used in this Report bohr =ao=atomic unit of lengths 0.529 x 10-10 m hartree=atomic unit of energy, e2/4 eoatomic unit ofelectric dipole =eao — 8.478 x 10 30Cm atomic unit of electric quadrupole=ea02 4.487 x 10 40 C m2 etc. 

C) In addition to the SI and cgs systems, we can define a system of Hartree48 atomic units (a.u.). Alas, a slightly different set of Rydberg49 a.u. also exists, but will not be discussed here. The Hartree atomic units are defined so that (i) the unit of length [L] is a0 = l bohr = 5.29177 x 10 11 m = radius of the "Bohr50 orbit" for hydrogen,... 

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

For small highly symmetric systems, like atoms and diatomic molecules, the Hartree-Fock equations may be solved by mapping the orbitals on a set of grid points. These are referred to as numerical Hartree-Fock methods. However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known functions. Any type of basis function may in principle be used expo ... 

We consider a system of N interacting electrons with the Hamiltonian (in Hartree atomic units, i.e., m = e = fi = 47re0 = 1)... 

Our main goal is that of calculating the properties of a given system consisting of M nuclei and N electrons. In principle, this can be achieved by solving the Schrodinger equation. In Hartree atomic units (me = e = 4 = h = 1) this equation is in the static, time-independent case... 

The matrix elements of interest here are of the forms ( (a, n) (/ , m)), (i/f(oe, n) V Vf(/ , m)), and n) T Vf(/ , m)), where V and T are, respectively, the Coulomb and kinetic-energy operators. For a four-body system, with respective charges qi,. ..,q4 and masses mi,m, with all quantities expressed in Hartree atomic units, these matrix elements can be written entirely in terms of the interparticle coordinates [12], with the potential-energy matrix elements given as... 

We consider a molecule with M nuclei and N electrons. The positions of the nuclei are denoted. i, Ri,, Rm and those of the electrons n, 2,. .., fjv. Moreover, we use Hartree atomic xmits and set accordingly nie= e = 4jt o = h=. The mass and charge of the Ath nucleus are then and Z, respectively. The combined coordinate jq denotes the position and spin coordinate of the rth electron. In the absence of external interactions and relativistic effects, the Hamilton operator for this system can then be written as a sum of five terms. 

For simplicity, we will write all formulae for spin-saturated systems. Obviously, spin can be easily included in all expressions when necessary. Hartree atomic units (e = ft = m = 1) will be used throughout this chapter. 

As in the nonrelativistic case, most of the salient features of the atomic systems are exposed in the treatment of the simplest of these, the hydrogen-like one-electron atoms. In Hartree atomic units the time-independent Dirac equation yields the coupled equations... 

In this book we have used two systems of units. The first is the SI system, which we use in the early chapters of the book and in particular for electromagnetic quantities. Factors of c therefore always represent the speed of light and never a conversion factor for magnetic units. The second is the Hartree atomic units system, defined by b = e = me = 1. In these units, c 137. Many physics texts use the system b = e = c = 1, since they are dealing with particles of different masses. Our concern is principally with the electron and chemistry, and the size of relativistic effects, which are measured by c, so Hartree atomic units are more appropriate. However, to keep the connection with SI units and to track quantities that involve the charge, the mass, or spin, the symbols ft, e, and m = me are retained in much of the development, whereas l/4 reo is usually omitted for clarity. 

We frequently express the operators in (1.1.2) in Hartree atomic units (see page xiv), the units of charge , mass and action then being e, m and ft respectively. The unit of length in this system is the Bohr radius flo = 5.292 X 10 m and the unit of energy is the Hartree = 27.21 eV. More precise values of these and many other constants have been compiled by Cohen and DuMond (1965). In atomic units. 

In the SCF MO theory the bond network is determined by the occupied MO in the system ground state. For reasons of simplicity we assume the closed-shell ( ) configuration of N = 2n electrons in the standard (spin-restricted) Hartree-Fock (RHF) description, which involves n lowest (doubly occupied, orthonormal) MO. In the familiar LCAO MO approach they are expanded as linear combinations of the (Lowdin) orthogonalized AO t = (Zi> Z2> > Zm) = iXi) crnitributed by the system constituent atoms ... 

This expression is not orbitally dependent. As such, a solution of the Hartree-Fock equation (equation (Al.3.18) is much easier to implement. Although Slater exchange was not rigorously justified for non-unifonn electron gases, it was quite successfiil in replicating the essential features of atomic and molecular systems as detennined by Hartree-Fock calculations. 

A simple example would be in a study of a diatomic molecule that in a Hartree-Fock calculation has a bonded cr orbital as the highest occupied MO (HOMO) and a a lowest unoccupied MO (LUMO). A CASSCF calculation would then use the two a electrons and set up four CSFs with single and double excitations from the HOMO into the a orbital. This allows the bond dissociation to be described correctly, with different amounts of the neutral atoms, ion pair, and bonded pair controlled by the Cl coefficients, with the optimal shapes of the orbitals also being found. For more complicated systems... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

As formulated above in terms of spin-orbitals, the Hartree-Fock (HF) equations yield orbitals that do not guarantee that P possesses proper spin symmetry. To illustrate the point, consider the form of the equations for an open-shell system such as the Lithium atom Li. If Isa, IsP, and 2sa spin-orbitals are chosen to appear in the trial function P, then the Fock operator will contain the following terms ... 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

Optimize these three molecules at the Hartree-Fock level, using the LANL2DZ basis set, LANL2DZ is a double-zeta basis set containing effective core potential (ECP) representations of electrons near the nuclei for post-third row atoms. Compare the Cr(CO)5 results with those we obtained in Chapter 3. Then compare the structures of the three systems to one another, and characterize the effect of changing the central atom on the overall molecular structure. 

---