Many-electron wave functions atomic orbitals approximation

Almost all approaches to many-electron wave functions, for both atoms and molecules, involve their formulation as products of one-electron orbitals. If the interelectronic repulsion term in (6.23) is small compared with the other terms, the Hamiltonian is approximately separable into independent operators for each electron and the two-electron wave function, f (1, 2), can be written as a simple product of one-electron functions,... 

In the most commonly utilized approximation, the many-electron wave functions are written in terms of products of one-electron wave functions similar to the solutions obtained for the hydrogen atom. These one-electron functions used to construct the many-electron wave function are called atomic orbitals. They are also called hydrogen-like orbitals since they are one-electron orbitals and also because their shape is similar to that of the hydrogen atom orbitals. 

We have placed special emphasis on using a consistent notation throughout the book. Since quantum chemists use a number of different notations, it is appropriate to define the notation we have adopted. Spatial molecular orbitals (with latin indices ij, k...) are denoted by These are usually expanded in a set of spatial (atomic) basis functions (with greek indices ju, V, A,...) denoted by 0. Molecular spin orbitals are denoted by x Occupied molecular orbitals are specifically labeled by a, b, c,... and unoccupied (virtual) molecular orbitals are specifically labeled by r, s, r,... Many-electron operators are denoted by capital script letters (for example, the Hamiltonian is Jf), and one-electron operators are denoted by lower case latin letters (for example, the Fock operator for electron-one is /( )). The exact many-electron wave function is denoted by O, and we use T to denote approximate many-electron wave functions (i.e., the Hartree-Fock ground state wave function is o while FS is a doubly excited wave function). Exact and approximate energies are denoted by S and , respectively. All numerical quantities (energies, dipole moments, etc.) are given in atomic units. 

Any theoretical method applicable to molecules may be also used for atoms, albeit very accurate wave functions, even for simple atoms, are not easy to calculate. In fact for atoms we know the solutions quite well only in the mean field approximation, i.e. the atomic orbitals. Such orbitals play an important role as building blocks of many-electron wave functions. 

A molecule or solid is a system build up from atomic nuclei and the corresponding number of electrons. With an appropriate Hamiltonian H, such system is described by a wave function 1, the solution of the eigenvalue problem // = . For fixed position of the nuclei (Bom-Oppenheimer approximation), the wave function is a 3Wdimensional mathematical object describing the movement of N electrons. The many-electron wave function is usually expressed as expansions using products of one-electron functions (orbitals) possibly expanded with chosen set of primitive functions (Slater functions, Gauss functions, plane waves, etc.). Even for time-independent nonrelativistic wave function, it is very difficult to perform an analysis of the system without further approximations or reductions. 

The Schrodinger equation can be solved approximately for atoms with two or more electrons. There are many solutions for the wave function, ij/, each associated with a set of numbers called quantum numbers. Three such numbers are given the symbols n, , and mi. A wave function corresponding to a particular set of three quantum numbers (e.g., n = 2, = 1, mi = 0) is associated with an electron occupying an atomic orbital. From the expression for ij/y we can deduce the relative energy of that orbital, its shape, and its orientation in space. 

The starting point of the creation of the theory of the many-electron atom was the idea of Niels Bohr [1] to consider each electron of an atom as orbiting in a stationary state in the field, created by the charge of the nucleus and the rest of the electrons of an atom. This idea is several years older than quantum mechanics itself. It allows one to construct an approximate wave function of the whole atom with the help of one-electron wave functions. They may be found by accounting for the approximate states of the passive electrons, in other words, the states of all electrons must be consistent. This is the essence of the self-consistent field approximation (Hartree-Fock method), widely used in the theory of many-body systems, particularly of many-electron atoms and ions. There are many methods of accounting more or less accurately for this consistency, usually named by correlation effects, and of obtaining more accurate theoretical data on atomic structure. 

Usually, the. spatial function ilt is constructed from the summation of one-electron spatial orbitals (atomic orbitals) 4>. known as the basis set. u.sed to construct a MO. This approach is known as the LCAO method (/inear combination of utomic orbitals). It is an approximation of the accurate many-elcetron wave function (Eq. 28-54). The atomic orbital contributions are weighted by coefficients c,. The summation is truncated, so the ip function is not complete, which has consequences when. solving for E. 

The LCAO method extends to molecules the description developed for many-electron atoms in Section 5.2. Just as the wave function for a many-electron atom is written as a product of single-particle AOs, here the electronic wave function for a molecule is written as a product of single-particle MOs. This form is called the orbital approximation for molecules. We construct MOs, and we place electrons in them according to the Pauli exclusion principle to assign molecular electron configurations. 

The theory of the chemical bond is one of the clearest and most informative examples of an explanatory phenomenon that probably occurs in some form or other in many sciences (psychology comes to mind) the semiautonomous, nonfundamental, fundamentally based, approximate theory (S ANFFBAT for short). Chemical bonding is fundamentally a quantum mechanical phenomenon, yet for all but the simplest chemical systems, a purely quantum mechanical treatment of the molecule is infeasible especially prior to recent computational developments, one could not write down the correct Hamiltonian and solve the Schrodinger equation, even with numerical methods. Immediately after the introduction of the quantum theory, systems of approximation began to appear. The Born Oppenheimer approximation assumed that nuclei are fixed in position the LCAO method assumed that the position wave functions for electrons in molecules are linear combinations of electronic wave functions for the component atoms in isolation. Molecular orbital theory assumed a characteristic set of position wave functions for the several electrons in a molecule, systematically related to corresponding atomic wave functions. 

Using the HMO approximation, the ir-electron wave function is expressed as a linear combination of atomic orbitals (for the case in which the plane of the molecule coincides with the y-z plane of the coordinate axis), in much the same way as described previously for the generalized MO method. Minimizing the total -jT-electron energy of the molecule with respect to the coefficients (the variation method) leads to a series of equations from which the coefficients can be extracted by way of a secular determinant. The mathematical operations involved in solving the equations are not difficult. We will not describe them in detail, but will instead concentrate on the interpretation of the results of the calculations. For many systems, the Huckel MO energies and atomic coefficients have been tabulated." ... 

The relativistic Hamilton operator for an electron can be derived, using the correspondence principle, from its relativistic classical Hamiltonian and this leads to the one-electron Dirac equation, which does contain spin operators. From the one-electron Dirac equation it seems trivial to define a many-electron relativistic equation, but the generalization to more electrons is less straightforward than in the non-relativistic case, because the electron-electron interaction is not unambiguously defined. The non-relativistic Coulomb interaction is often used as a reasonable first approximation. The relativistic treatment of atoms and molecules based on the many-electron Dirac equation leads to so-called four-component methods. The name stems from the fact that the electronic wave functions consist of four instead of two components. When the couplings between spin and orbital angular moment are comparable to the electron-electron interactions this is the preferred way to explain the electronic structure of the lowest states. 

The electron configuration is essentially a first-guess, inaccurate spatial wavefimction for the many-electron atom. It is generated by the product of a number of one-electron wavefunctions, the atomic orbitals. We can write this wave-function explicitly, but we must remember that the coordinates of each electron are distinct. Knowing that each spatial wavefimction depends on the r, d, and 0 coordinates for that electron, let s now just write 4fn.i,m,W to indicate the spatial wavefimction for electron 1, a function of the coordinates r, 6i, and For example, the spatial part of the wavefimction for the Li atom configuration with two electrons in the Is orbital and one in the 2s could be approximated ... 

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