Electron orbital

In quantum chemistry, an electron orbital (or more often just orbital) is a synonym for a quadratically integrable one-electron wave function. Here "orbital" is used as a noun. In quantum mechanics, the adjective orbital is often used as a synonym of "spatial", in contradistinction to spin.

Definitions of orbitals

Several kinds of orbitals can be distinguished.

Atomic orbital

The basic kind of orbital is the atomic orbital (AO). This 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, but form the basis of many qualitative arguments in chemical bonding and atomic spectroscopy.

The orbital is quadratically integrable, 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.

Molecular orbital

The second kind of orbital is the molecular orbital (MO). Such a one-electron function depends on several vectors: r_A1, r_B1, r_C1, ... where A, B, C, ... are different points in space (usually nuclear positions). The oldest example of an MO (without use of the name MO yet) is in the work of Burrau (1927) on the single-electron ion H₂⁺. Burrau applied spheroidal coordinates (a bipolar coordinate system) to describe the wavefunctions of the electron of H₂⁺.

Lennard-Jones introduced the following linear combination of atomic orbitals (LCAO) way of writing an MO φ:

$\phi (\mathbf {r} _{1})=\sum _{A=1}^{N_{\mathrm {nuc} }}\sum _{i=1}^{n_{A}}c_{Ai}\chi _{i}(\mathbf {r} _{A1}),\qquad c_{Ai}\in \mathbb {C} ,$

where A runs over N_nuc different points in space (usually A runs over all the nuclei of a molecule, hence the name molecular orbital), and i runs over the n_A different AOs centered at A. The complex coefficients c _Ai can be calculated by any of the existing effective-one-electron quantum chemical methods. Examples of such methods are the Hückel method and the Hartree-Fock method.

Spinorbitals

The AOs and MOs defined so far depend only on the spatial coordinate vector r_A1 of electron 1. In addition, an electron has a spin coordinate μ, which can have two values: spin-up or spin-down. A complete set of functions of μ consists of two functions only, traditionally these are denoted by α(μ) and β(μ). These functions are eigenfunctions of the z-component s_z of the spin angular momentum operator with eigenvalues ±½.

Spin atomic orbital

The most general spin atomic orbital of electron 1 is of the form

$\chi _{i}^{+}(\mathbf {r} _{A1})\alpha (\mu _{1})+\chi _{i}^{-}(\mathbf {r} _{A1})\beta (\mu _{1})$

which in general is not an eigenfunction of s_z. More common is the use of either

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

which are eigenfunctions of s_z. Since it is rare that different AOs are used for spin-up and spin-down electrons, we dropped the superscripts + and −.

Spin molecular orbital

A spin molecular orbital is usually either

$\phi ^{+}(\mathbf {r} _{1})\alpha (\mu _{1})\quad {\hbox{or}}\quad \phi ^{-}(\mathbf {r} _{1})\beta (\mu _{1}).$

Here the superscripts + and − might be necessary, because some quantum chemical methods distinguish the spatial wave functions of electrons with different spins. These are the so-called different orbitals for different spins (DODS) (or spin-unrestricted) methods. However, many quantum chemical methods apply the spin-restriction:

$\phi ^{+}(\mathbf {r} _{1})=\phi ^{-}(\mathbf {r} _{1})=\phi (\mathbf {r} _{1}).$

Chemists express this spin-restriction by stating that two electrons [electron 1 with spin up (α), and electron 2 with spin down (β)] are placed in the same spatial orbital φ. This means that the total N-electron wave function contains a factor of the type φ(r₁)α(μ₁)φ(r₂)β(μ₂).

History of term

Orbit is an old noun (1548), initially indicating the path of the Moon and later the paths of other heavenly bodies as well. The adjective "orbital" had (and still has) the meaning "relating to an orbit". When Ernest Rutherford in 1911 postulated his planetary model of the atom (the nucleus as the Sun, and the electrons as the planets) it was natural to call the paths of the electrons "orbits". Bohr, although he was the first to recognize (1913) orbits as stationary states of the hydrogen atom, used the word as well. However, after Schrödinger (1926) had solved his wave equation for the hydrogen atom (see this article for details), it became clear that the electronic "orbits" did not resemble planetary orbits at all. The wave functions of the hydrogen electron are time-independent and smeared out. They are more like unmoving clouds than like planetary orbits. As a matter of fact, the angular parts of the hydrogen wave functions are spherical harmonics and hence they have the same appearance as spherical harmonics. (See spherical harmonics for a few graphical illustrations).

In the 1920s electron spin was discovered, whereupon the adjective "orbital" started to be used in the meaning of "non-spin", that is, as a synonym of "spatial". In scientific papers of around 1930 one finds discussions about "orbital degeneracy", meaning that the spatial (non-spin) parts of several one-electron wave functions have the same energy. Also the terms orbital- and spin-angular momentum date form these days.

In 1932 Robert S. Mulliken coined the noun "orbital". He wrote: From here on, one-electron orbital wave functions will be referred to for brevity as orbitals. Then he went on to distinguish atomic and molecular orbitals.

Later the somewhat unfortunate term "spinorbital" was introduced for the product functions φ(r₁)α(μ₁) and φ(r₁)β(μ₁) in which φ(r₁) has the tautological name "spatial orbital" and α(μ₁) and β(μ₁) are called "one-electron spin functions". The term "spinorbital" is unfortunate because it merges in one word the concepts of spin and orbital, which were distinguished carefully by early writers on quantum mechanics. For instance, one of the pioneers of theoretical chemistry, Walter Heitler, correctly contraposes two-electron spin functions and two-electron orbital functions. In the phrase two-electron orbital function, Heitler uses orbital as an adjective synonymous with spatial (non-spin). Note, parenthetically, that Heitler does not refer to a "two-electron orbital", (there is no such thing as a two-electron orbital!), but that an inexperienced reader may easily misread the term "two-electron orbital function" as "two-electron orbital".

Applications

Before the advent of electronic computers, orbitals (molecular as well as atomic) were used extensively in qualitative arguments explaining several properties of atoms and molecules. Orbitals still play this role in introductory texts and also in organic chemistry, where orbitals serve in the explanation of some reaction mechanisms, for instance in the Woodward-Hoffmann rules. In modern computational quantum chemistry the role of atomic orbitals is different; they serve as a convenient expansion basis, comparable to powers of x in a Taylor expansion of a function f(x), or sines and cosines in a Fourier series.

Atomic orbitals

Bohr was the first to see how atomic orbitals form a basis for an understanding of the Periodic Table of elements. In the explanation of the Periodic Table, atomic orbitals are labeled as if they are hydrogen orbitals, that is, by a principal quantum number n and a letter designating the azimuthal quantum number (angular momentum quantum number) l. This labeling is not only correct for hydrogen-like orbitals, but also for AOs that arise as solutions from an N-electron independent particle model with a central (spherically symmetric) potential field. A central field is necessary to have l as a good quantum number, and the concept "orbital" is tied to an independent particle model, such as the Hartree and the Hartree-Fock model.

In an independent particle model an Ansatz for the total N-electron wave function is a (possibly antisymmetrized) product of orbitals. From the effective-one-electron Schrödinger equation ensuing from this Ansatz, the actual forms of the orbitals and orbital energies are obtained, see Hartree-Fock method for an example and more details. Two major differences between such an effective-one-electron model and the analytic solution of the hydrogen atom are: (i) The radial functions are given numerically, and are no longer known analytic functions, such as the Laguerre functions for the hydrogen atom. The angular parts, however, are the same analytic functions as for the H-atom (spherical harmonics). (ii) The orbitals within a given atomic shell (orbitals with same pricipal quantum number n) are no longer degenerate (have no longer the same energy). While in the hydrogen atom, for instance the orbitals in the n = 3 shell: 3s, 3p, and 3d, all have the same energy (proportional to 1/n² = 1/3²), the degeneracy for the N-electron atom is lifted: only those of same azimuthal quantum number l are degenerate. The principal quantum number is now simply a counting index, it counts in increasing energy orbitals of the same l, starting at n = l + 1 (to be in line with the hydrogen AOs). It is found that the orbitals arising from the central field, independent particle model 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.

Remark: for some atoms the order of 3d and 4s is flipped and for some other atoms the order of 4d and 5s is flipped.

One can now 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 priciple that forbids more than two electrons per spatial orbital. Allowed is at most one electron with α spin and one with β spin. Recall further that there are 2l+1 orbitals of definite l. For instance, the neon atom (atomic number Z = 10) has the electronic configuration (n = 1 and n = 2 shells are completely filled):

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

while 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.

Molecular orbitals

In the hands of Friedrich Hund, Robert S. Mulliken, John Lennard-Jones, and others, molecular orbital theory was established firmly in the 1930s as a (mostly qualitative) theory explaining much of chemical bonding, especially the bonding in diatomic molecules.

MO energy level diagram for H₂

To give the flavor of the theory we look at the simplest molecule: H₂. The two hydrogen atoms, labeled A and B, each have one electron (a red arrrow in the figure) in an 1s atomic orbital (AO). These AOs are labeled in the figure 1s_A and 1s_B. When the atoms start to interact the AOs combine linearly to two molecular orbitals:

$\sigma _{g}=N_{g}(1s_{A}+1s_{B})\quad {\hbox{and}}\quad \sigma _{u}=N_{u}(1s_{A}-1s_{B}).$

Earlier in this article a general expression for an LCAO-MO was given. In the present example of the σ_g MO of H₂, we have N_nuc = 2, n_A = n_B = 1, χ_A1 = 1s_A, χ_B1 = 1s_B, c_A1 = c_B1 = N_g. With regard to the form of the MOs the following: Because both hydrogen atoms are identical, the molecule has reflection symmetry with respect to a mirror plane halfway the H—H bond and perpendicular to it. It is one of the basic assumptions in quantum mechanics that wave functions show the symmetry of the system. [Technically: solutions of the Schrödinger equation belong to a subspace of Hilbert (function) space that is irreducible under the symmetry group of the system]. Clearly σ_g is symmetric (is even, stays the same) under

$\mathrm {\scriptstyle Reflection:} \quad 1s_{A}\longleftrightarrow 1s_{B},$

while σ_u changes sign (is antisymmetric, is odd). The Greek letter σ indicates invariance under rotation around the bond axis. The subscripts g and u stand for the German words gerade (even) and ungerade (odd). So, because of the high symmetry of the molecule we can immediately write down two molecular orbitals, which evidently are linear combinations of atomic orbitals.

The normalization factors N_g and N_u and the orbital energies are still to be computed. The normalization constants follow from requiring the MOs to be normalized to unity. In the computation we will use the bra-ket notation for the integral over x_A1, y_A1, and z_A1 (or over x_B1, y_B1, and z_B1 when that is easier).

$1=\langle \sigma _{g}|\sigma _{g}\rangle =N_{g}^{2}\left(\langle 1s_{A}|1s_{A}\rangle +\langle 1s_{B}|1s_{B}\rangle +\langle 1s_{A}|1s_{B}\rangle +\langle 1s_{B}|1s_{A}\rangle \right)=N_{g}^{2}(2+2S).$

We used here that the AOs are normalized to unity and that the overlap integral is real

$\langle 1s_{A}|1s_{B}\rangle =\langle 1s_{B}|1s_{A}\rangle \equiv S.$

Applying the same procedure for N_u we find

$N_{g}=\left[2+2S\right]^{-1/2}\quad {\hbox{and}}\quad N_{u}=\left[2-2S\right]^{-1/2}.$

At this point one often assumes that S ≅ 0, so that both normalization factors are equal to ½√2.

In order to calculate the energy of an orbital we introduce an effective-one-electron energy operator (Hamiltonian) h,

${\begin{aligned}E_{\sigma _{g}}&=\langle \sigma _{g}|h|\sigma _{g}\rangle \\&={\frac {1}{2}}\left(\langle 1s_{A}|h|1s_{A}\rangle +\langle 1s_{B}|h|1s_{B}\rangle +\langle 1s_{A}|h|1s_{B}\rangle +\langle 1s_{B}|h|1s_{A}\rangle \right)\\&\equiv {\frac {1}{2}}(q_{A}+q_{B}+\beta _{AB}+\beta _{BA})=q+\beta .\end{aligned}}$

Since A and B are identical atoms equipped with identical AOs, we were allowed to use

$q_{A}=q_{B}=q=\langle 1s|h|1s\rangle =E_{1s}\quad {\hbox{and}}\quad \beta _{AB}=\beta _{BA}=\beta .$

Likewise

$E_{\sigma _{u}}=\langle \sigma _{u}|h|\sigma _{u}\rangle =q-\beta .$

The energy term β is negative (causes the attraction between the atoms). One may tempted to assume that q = −½ hartree (the energy of the 1s orbital in the free atom). This is not the case, however, because h contains the attraction with both nuclei, so that q is distance dependent.

If we now look at the figure in which energies are increasing in vertical direction, we see that the two AOs are at the same energy level q, and that the energy of σ_g is a distance |β| below this level, while the energy of σ_u is a distance |β| above this level. In the present simple-minded effective-one-electron model we can simple add the orbital energies and find that the bonding energy in H₂ is 2β (two electrons in the bonding MO σ_g, ignoring the distance dependence of q). This model predicts that the bonding energy in the one-electron ion H₂⁺ is half that of H₂, which is correct within a 20% margin.

If we apply the model to He₂, which has four electrons, we find that we must place two electrons in the bonding orbital and two electrons in the antibonding MO σ_u, with the total energy being 2β − 2β = 0. So, this simple application of molecular orbital theory predicts that H₂ is bound and that He₂ is not, which is in agreement with the observed facts.

Atomic orbital basis sets

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

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, but computer time limits the number in practice. (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, ${\scriptstyle L^{2}[\mathbb {R} ^{3}]}$, 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 boron atom with 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 applying minimum basis sets. However, it soon became clear that such a basis set gave very disappointing results. After this discovery, in the late 1960s and early 1970s, construction of good AO basis became an important subject of research.

In order to explain what such a construction comprises, we must first refer to the article Gauss type orbitals (GTOs), because present-day computer programs are using almost exclusively GTOs. In that article some required terminology is introduced. In a minimum GTO basis set (also known as a single zeta basis set) every atomic orbital (and its degenerate partners) occupied in the free atom is represented by a single contracted set of GTOs. The term "single zeta" is historic and refers back to the days that Slater type orbitals were used universally and to the fact that the screening constant in an STO is conventionally indicated by the Greek letter zeta (ζ).

(to be continued)

References and notes

(To be continued)