Atomic orbital

In quantum mechanics and quantum chemistry, an atomic orbital is a function of an electron that describes the motion—in the quantum mechanical sense of the word—of the electron around the nucleus of an atom.

Originally an atomic orbital (AO) was seen as a solution of a one-electron Schrödinger equation for an atomic electron. In the case of one-electron atoms this Schrödinger equation is exact and its solutions are known in analytic form. In the case of more-electron atoms the Schrödinger equation is not solvable and is replaced by an effective one-electron equation that contains the interaction between the electrons in an averaged manner. Since formally the effective equation depends only on the coordinates x, y, z of one given electron, its solution is a function of x, y, and z—an atomic orbital. If furthermore the effective Schrödinger equation is invariant under rotation, as is the case for the hydrogen atom, it can be shown that the angular parts of the solutions consist of spherical harmonic functions. The latter functions determine the shape of the AOs.

Later, especially in computationally oriented branches of quantum chemistry, AOs simply became synonomous with atomic one-electron functions. These functions are used as basis functions in calculations that compute linear combinations of AOs (LCAOs) as solutions of an effective one-electron Schrödinger equation. It is common to choose solid harmonic functions (which differ from spherical harmonics only by a factor r ^l ) as the angular parts of the basis functions.

Formal definition of AOs

An atomic orbital is a function depending on a single 3-dimensional vector r_A1, which is a vector pointing from point A to electron 1. Generally there is a nucleus at A. The following notations for an AO are frequently used,

$\chi _{i}(\mathbf {r} _{A1}),\quad {\hbox{or}}\quad \chi _{Ai}(\mathbf {r} _{1})$

but other notations can be found in the literature. In the second notation the center A is added as an index to the orbital. We say that χ_{A i} (or, as the case may be, χ _i) is centered at A. In numerical computations AOs are either taken as Slater type orbitals (STOs) or Gaussian type orbitals (GTOs). Hydrogen-like orbitals are rarely applied in numerical calculations, because they are not complete and fairly difficult to handle.

The orbital is quadratically integrable (has finite norm), which means that the following integral is finite,

$0\leq \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\chi _{i}(\mathbf {r} _{A1})^{*}\chi _{i}(\mathbf {r} _{A1})\;dx_{A1}\,dy_{A1}\,dz_{A1}\ <\infty .$

Its integrand being real and non-negative, the integral is real and non-negative. The integral is zero if and only if χ _i is the zero function.

Solutions of the atomic Schrödinger equation

Bohr was the first to see how atomic orbitals form a basis for an understanding of the Periodic Table of elements. In Bohr's explanation of the Periodic Table, the atomic orbitals carry the same labels as hydrogen orbitals. That is, they have a principal quantum number n and a letter (s, p,...) designating the azimuthal quantum number (angular momentum quantum number) l.

The quantum numbers n and l are also valid labels for AOs that arise as solutions from an N-electron independent-particle model with a central (spherically symmetric) potential field. The central field approximation, which entails the spherical averaging of the field due to the electrons, is necessary to have l as a good quantum number. An independent-particle model leads to an effective one-electron Schrödinger equation—one electron in the averaged (mean) field of the other electrons. The solutions of a mean field equation are orbitals and orbital-energies. So, orbitals and orbital-energies do not only arise in real one-electron systems, such as the H-atom, but also in independent-particle approximations of N-electron systems.

Two major differences between an atomic effective one-electron model and the hydrogen-like atom are:

1. The radial solutions are given numerically, and are no longer known analytic functions, such as the Laguerre functions for the hydrogen-like atom. The angular parts, however, are the same analytic functions (spherical harmonics) as in the case of the hydrogen-like-atom.
2. The high degeneracy of the H-atom is lifted. In the hydrogen atom all orbitals of certain n are degenerate. For instance, the orbitals in the n = 4 shell: 4s, 4p, 4d, and 4f, all have the same energy (proportional to 1/n² = 1/16). In the N-electron atom only the AOs of the same azimuthal quantum number l are degenerate.

Note the role of the principal quantum number n in the effective one-electron model. It is simply an index that counts increasing orbital energies, starting at n = l + 1 (to be in line with the hydrogen-like AOs). In summary, an atomic orbital of an N-electron atom is labeled by n (indicating its energy) and l (indicating its angular momentum).

It is found that the orbitals arising from the central field, independent-particle model applied to different atoms have the following order in increasing energy

1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 5s, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d.

For some atoms the order of 3d and 4s is flipped and for some other atoms the order of 4d and 5s is flipped. Note further that, for example 4f is not degenerate with 4d, which is the case for hydrogen-like atoms.

Knowing this order, one can build up the atoms in the Periodic Table by the Aufbau (building-up) principle: fill orbitals in increasing energy. Doing this one must obey the Pauli exclusion principle that forbids more than two electrons per spatial orbital. Allowed is at most one electron with α spin and one with β spin. In addition recall that there are 2l+1 orbitals of certain l. For instance, the neon atom (atomic number Z = 10) has the electronic configuration (n = 1 and n = 2 shells are completely filled):

${\scriptstyle 1s^{2}\,2s^{2}\,2p^{6},}$

meaning two electrons are in the 1s, two electrons in the 2s, and six electrons in the 2p AOs. Similarly, chlorine (Z = 17) has

${\scriptstyle 1s^{2}\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{5}\,}.$

For more details the article Periodic Table may be consulted. A list of the electronic configurations of the ground states of the first 94 elements may be found here.

AO basis sets

As stated above, in computational quantum chemistry atomic orbitals are used as a mathematical device to obtain good approximations of N-electron atomoc and molecular wave functions—solutions of the time-independent electronic Schrödinger equation.

A computation of a molecular wave function usually goes through the following steps:

1. A geometry of the molecule is chosen (in accordance with the Born-Oppenheimer approximation the nuclei are clamped in space).
2. A set of AOs is chosen which are centered on the nuclei. Sometimes the AO set is augmented with orbitals in the middle of bonds, where there are no nuclei. Preferably the AO basis set is as close as possible to a complete basis of one-electron Hilbert space, ${\scriptstyle L^{2}[\mathbb {R} ^{3}]}$, but computer time is a practical limit. (Many methods require computer times proportional to n⁶ or n⁷, where n is the number of AOs.)
3. An LCAO Hartree-Fock calculation yields the MO coefficients c_{A i} and the same number of MOs as AOs (namely n).
4. The MOs are used in a post-Hartree-Fock calculation (configuration interaction, Møller-Plesset perturbation theory, coupled cluster theory, etc.).

The AOs and MOs spanning the very same orbital subspace of one-electron Hilbert space, it would be conceivable to skip the Hartree-Fock calculation. However, it turns out that the post-Hartree-Fock methods converge much better when they are based on MOs instead of on (orthogonalized) AOs.

The size of an AO basis is of crucial importance. The qualitative, pre-computer, MO-theoretical studies were invariably based on minimum basis sets. That is, only orbitals occupied in the free atoms were included in the basis. (But note that, for instance for the ground state boron atom with its electron configuration 1s²2s²2p, it cannot be said whether 2p_x, 2p_y, or 2p_z is occupied. In such a case all three degenerate p orbitals are included in the minimum basis set). It was natural that the first computer calculations followed this pattern and applied minimum basis sets. However, it soon was found that such basis sets give very disappointing results. After this became clear in the late 1960s and early 1970s, search for good AO basis became an important subject of research.

For the mathematical definition of basis sets, we refer to the article Gauss type orbitals (GTOs), because the majority of present-day computer programs handle only GTOs. In the GTO article the concepts of primitive orbital and contracted set are introduced. In a minimum GTO basis set (also known as a single-zeta basis set) every atomic orbital occupied in the free atom is represented by a single contracted set. The term "single-zeta" is historic and refers back to the days that Slater type orbitals were universally used and to the fact that the screening constant in an STO is conventionally indicated by the Greek letter zeta (ζ). Single-zeta (SZ) basis sets giving poor quantitative results, the next step is the use of double-zeta (DZ) basis set, which involves a doubling of the SZ basis. Triple-zeta (TZ) (tripling of the minimum basis), quadruple-zeta (QZ), quintuple-zeta (5Z), sextuple-zeta (6Z) basis sets all have been proposed and have been constructed.

For instance a QZ GTO basis for HCN includes : 4 s-orbitals (s-type contracted sets) on H, 8 s-orbitals on both C and N, 4 p_x-, 4 p_y-, and 4 p_z-orbitals on C and N. In all these cases the construction of the contracted set (AO) involves the determination of the exponents of the primitive Gaussians and the corresponding contraction coefficients.

It is known that an atom polarizes (gets a dipole moment) under influence of an external electric field. It is also known that, in the presence of such a field, AOs of higher l quantum numbers must be included in the basis in order to obtain reliable results. When, for instance, the ground state boron atom (1s²2s²2p) is polarized, there will be a mixing, linear in the strength of the external field, between 2p and 3d. That is, to describe the polarization of the boron atom correctly, one or more 3d sets (sets of five degenerate AOs) must be added to the basis of boron. Since in molecules strong electric fields, due to the nuclei, are present, it stands to reason that polarization functions improve the computational results. For an atom that only has occupied s orbitals, like hydrogen and helium, polarization functions start at l = 1. For atoms such as boron with occupied p-orbitals, the polarization functions are by definition d, f, g, etc. orbitals.

As a final conclusion it must be pointed out that the present-day elaborate AO bases do not have the physical meaning that was attached to AOs in precomputer times. The AOs are now looked upon as building bricks in a numerical approximation, no chemical of physical conclusions are to be drawn from the AOs themselves. The relevant conclusions about nature emerge from the final computational results. Especially in the post Hartree-Fock methods, the contribution of the AOs to the conclusions is deeply hidden in the output and not at all transparent.