imported>Paul Wormer |
imported>Henry A. Padleckas |
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| 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]]. | | In [[quantum chemistry]], an '''electron orbital''' (or more often just '''orbital''') is a synonym for a one-electron function, i.e., a function of a single vector '''r''', the position vector of the electron.<ref> Here "orbital" is used as a ''noun''. In [[quantum mechanics]], the ''adjective'' orbital is often used as a synonym of "spatial" (as in orbital [[angular momentum]]), in contrast to [[spin]] (as in spin angular momentum).</ref> |
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| | The majority of quantum chemical methods expect that an orbital has a finite [[norm]], i.e., that the orbital is normalizable (quadratically integrable), and hence this requirement is often added to the definition. |
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| | In other branches of chemistry, an orbital is often seen as a [[wave function]] of an electron, meaning that an orbital is seen as a solution of an (effective) ''one-electron [[Schrödinger equation]]''. This point of view is a narrowing of the more general quantum chemical definition, but not contradictory to it. In the past quantum chemists, too, distinguished one-electron functions from orbitals; one-electron functions were fairly arbitrary functions of the position vector '''r''', while orbitals were solutions of certain effective one-electron Schrödinger equations. This distinction faded out in quantum chemistry, resulting in the present definition of "normalizable one-electron function", not necessarily eigenfunctions of a one-electron [[Hamiltonian]]. |
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| | Usually one distinguishes two kinds of orbitals: ''atomic orbitals'' and ''molecular orbitals''. |
| | Atomic orbitals are expressed with respect to one Cartesian system of axes centered on a single atom. Molecular orbitals (MOs) are "spread out" over a molecule. Usually this is a consequence of an MO being a linear combination (weighted sum) of atomic orbitals centered on different atoms. |
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| ==Definitions of orbitals== | | ==Definitions of orbitals== |
| Several kinds of orbitals can be distinguished. | | Several kinds of orbitals can be distinguished. |
| ===Atomic orbital=== | | ===Atomic orbital (AO)=== |
| The basic kind of orbital is the ''atomic orbital'' (AO). This is a function depending on a single 3-dimensional vector '''r'''<sub>''A''1</sub>, which is a vector pointing from point ''A'' to electron 1. Generally there is a nucleus at ''A''.<ref>Floating AOs and bond functions, both of which have an empty point ''A'', are sometimes used.</ref>
| | ''See the article [[atomic orbital]]'' |
| The following notations for an AO are frequently used,
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| :<math>
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| \chi_i(\mathbf{r}_{A1}),\quad\hbox{or}\quad\chi_{Ai}(\mathbf{r}_1)
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| </math>
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| 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<sub>''A i''</sub> (or, as the case may be, χ<sub>'' i''</sub>) is ''centered at'' ''A''. In numerical computations AOs are either taken as [[Slater orbital| Slater type orbitals]] (STOs) or [[Gaussian type orbitals]] (GTOs). [[Hydrogen-like atom|Hydrogen-like orbitals]] are rarely applied in numerical calculations, but form the basis of many qualitative arguments in chemical bonding and atomic spectroscopy.
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| The orbital is quadratically integrable, which means that the following integral is finite,
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| :<math>
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| 0 \le \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.
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| </math>
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| Its integrand being real and non-negative, the integral is real and non-negative. The integral is zero if and only if χ<sub>'' i''</sub> is the zero function.
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| ===Molecular orbital=== | | ===Molecular orbital (MO)=== |
| The second kind of orbital is the ''molecular orbital'' (MO). Such a one-electron function depends on several vectors: '''r'''<sub>''A''1</sub>, '''r'''<sub>''B''1</sub>, '''r'''<sub>''C''1</sub>, ... 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 H<sub>2</sub><sup>+</sup>. Burrau applied [[spheroidal coordinates]] (a bipolar coordinate system) to describe the wavefunctions of the electron of H<sub>2</sub><sup>+</sup>.
| | ''See the article [[molecular orbital]]'' |
| | |
| Lennard-Jones<ref>J. E.Lennard-Jones, ''The Electronic Structure of some Diatomic Molecules;; Trans. Faraday Soc. vol '''25''', p. 668 (1929).</ref> introduced the following ''linear combination of atomic orbitals'' (LCAO) way of writing an MO φ:
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| :<math>
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| \phi(\mathbf{r}_1) = \sum_{A=1}^{N_\mathrm{nuc}} \sum_{i=1}^{n_A} c_{Ai} \chi_i(\mathbf{r}_{A1}),
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| \qquad c_{Ai} \in \mathbb{C},
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| </math>
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| where ''A'' runs over ''N''<sub>nuc</sub> 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''<sub>''A''</sub> different AOs centered at ''A''. The complex coefficients ''c''<sub>'' Ai''</sub> 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]].
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| ===Spinorbitals=== | | ===Spinorbitals=== |
| The AOs and MOs defined so far depend only on the ''spatial coordinate vector'' '''r'''<sub>''A''1</sub> 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''<sub>''z''</sub> of the [[spin angular momentum operator]] with eigenvalues ±½. | | The AOs and MOs defined so far depend only on the ''spatial coordinate vector'' '''r''' of a single electron. 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''<sub>''z''</sub> of the [[Angular momentum (quantum)#Spin angular momentum|spin angular momentum operator]] with eigenvalues ±½. |
| ====Spin atomic orbital==== | | ====Spin atomic orbital==== |
| The most general ''spin atomic orbital'' of electron 1 is of the form | | The most general ''spin atomic orbital'' of an electron is of the form |
| :<math> | | :<math> |
| \chi_i^{+}(\mathbf{r}_{A1})\alpha(\mu_1) + \chi_i^{-}(\mathbf{r}_{A1})\beta(\mu_1) | | \chi_i^{+}(\mathbf{r})\alpha(\mu) + \chi_i^{-}(\mathbf{r})\beta(\mu), |
| </math> | | </math> |
| which in general is ''not'' an eigenfunction of ''s''<sub>''z''</sub>. More common is the use of either
| | where '''r''' is a vector from the nucleus of the atom to the position of the electron. |
| | In general this function is ''not'' an eigenfunction of ''s''<sub>''z''</sub>. More common is the use of either |
| :<math> | | :<math> |
| \chi_i(\mathbf{r}_{A1})\alpha(\mu_1) \quad\hbox{or}\quad \chi_i(\mathbf{r}_{A1})\beta(\mu_1), | | \chi_i(\mathbf{r})\alpha(\mu) \quad\hbox{or}\quad \chi_i(\mathbf{r})\beta(\mu), |
| </math> | | </math> |
| which are eigenfunctions of ''s''<sub>''z''</sub>. Since it is rare that different AOs are used for spin-up and spin-down electrons, we dropped the superscripts + and −. | | which are eigenfunctions of ''s''<sub>''z''</sub>. Since it is rare that different AOs are used for spin-up and spin-down electrons, we dropped the superscripts + and −. |
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| A ''spin molecular orbital'' is usually either | | A ''spin molecular orbital'' is usually either |
| :<math> | | :<math> |
| \phi^{+}(\mathbf{r}_{1})\alpha(\mu_1) \quad\hbox{or}\quad \phi^{-}(\mathbf{r}_{1})\beta(\mu_1) . | | \phi^{+}(\mathbf{r})\alpha(\mu) \quad\hbox{or}\quad \phi^{-}(\mathbf{r})\beta(\mu) . |
| </math> | | </math> |
| 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: | | 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: |
| :<math> | | :<math> |
| \phi^{+}(\mathbf{r}_{1}) = \phi^{-}(\mathbf{r}_{1}) = \phi(\mathbf{r}_{1}). | | \phi^{+}(\mathbf{r}) = \phi^{-}(\mathbf{r}) = \phi(\mathbf{r}). |
| </math> | | </math> |
| 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('''r'''<sub>1</sub>)α(μ<sub>1</sub>)φ('''r'''<sub>2</sub>)β(μ<sub>2</sub>). | | Chemists express this spin-restriction by stating that two electrons [an electron with spin up (α), and another electron 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('''r'''<sub>1</sub>)α(μ<sub>1</sub>)×φ('''r'''<sub>2</sub>)β(μ<sub>2</sub>). |
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| ==History of term==
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| 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".
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| 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 [[hydrogen-like atom|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).
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| 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.
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| In 1932 [[Robert S. Mulliken]]<ref>R. S. Mulliken, ''Electronic Structures of Molecules and Valence. II General Considerations'', Physical Review, vol. '''41''', pp. 49-71 (1932) </ref> coined the ''noun'' "orbital". He wrote: ''From here on, one-electron orbital wave functions will be referred to for brevity as orbitals''.<ref>Note that here, evidently, Mulliken uses the adjective "orbital" in the meaning of "spatial" (non-spin) and defines an orbital as, what is now called a "spatial orbital".</ref> Then he went on to distinguish atomic and molecular orbitals.
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| Later the somewhat unfortunate term "spinorbital" was introduced for the product functions φ('''r'''<sub>1</sub>)α(μ<sub>1</sub>) and φ('''r'''<sub>1</sub>)β(μ<sub>1</sub>) in which φ('''r'''<sub>1</sub>) has the tautological name "spatial orbital" and α(μ<sub>1</sub>) and β(μ<sub>1</sub>) 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''.<ref>W. Heitler, ''Elementary Wave Mechanics'', 2nd edition (1956) Clarendon Press, Oxford, UK.</ref> 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!). However, an inexperienced reader may easily and erroneously interpret the term "two-electron orbital function" as "two-electron orbital".
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| ==Applications== | | ==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]]. | | Before the advent of electronic computers, orbitals (molecular as well as atomic) were used extensively in qualitative arguments explaining all kinds of 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. | | 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. |
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| ===Atomic orbitals=== | | ===Atomic orbitals=== |
| Bohr<ref>N. Bohr, Zeitschrift für Physik, vol. '''9''', p. 1 (1922)</ref> was the first to see
| | Originally AOs were defined as approximate solutions of atomic Schrödinger equations, |
| how atomic orbitals form a basis for an understanding of the [[Periodic Table]] of elements. In Bohr's explanation of the Periodic Table, atomic orbitals carry the same labels as [[hydrogen-like atom|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''. This (''n'', ''l'') labeling is valid not only for hydrogen-like orbitals, but for other kinds of atomic orbitals as well. Indeed, ''n'' and ''l'' are 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 is necessary to have ''l'' as a good quantum number. Orbitals and orbital-energies do not only arise in one-electron systems, but also in independent-particle approximations to ''N''-electron systems, such as the Hartree and the [[Hartree-Fock]] models. The natural number ''n'' counts in these cases the orbital-energies.
| | see the section ''solution of the atomic Schrödinger equation'' in [[atomic orbital]] for more on this. Very often atomic orbitals are depicted in [[polar plots]]. In fact, the angular parts (functions depending on the [[spherical polar coordinates|spherical polar angles]] θ and φ) are commonly plotted. The angular parts of atomic orbitals are functions known as [[spherical harmonics]]. |
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| In an independent-particle model an ''Ansatz'' for the total ''N''-electron wave function is a (possibly [[antisymmetrizer|antisymmetrized]]) product of orbitals. The solution of the effective-one-electron Schrödinger equation, ensuing from this Ansatz, gives the orbitals and their energies, see [[Hartree-Fock method]] for an example and more details. Two major differences between such an effective-one-electron model and the hydrogen-like atom (a real one-electron system) are: (i) 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 as for the hydrogen-like-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: 3''s'', 3''p'', and 3''d'', all have the same energy (proportional to 1/''n''<sup>2</sup> = 1/3<sup>2</sup>), the degeneracy for the ''N''-electron atom is lifted: only those of same azimuthal quantum number ''l'' are degenerate. The principal quantum number in the effective-one-electron model is 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-like AOs).
| | As stated, quantum chemists see AOs simply as convenient basis functions for quantum mechanical computations. See the section ''AO basis sets'' in the article [[atomic orbital]] for more on the use of atomic orbitals in quantum chemistry. See [[Slater orbital]] for the explicit analytic form of orbitals and see [[Gauss type orbitals]] for a discussion of the type of AOs that are most frequently used in computations. |
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| It is found<ref>J. C. Slater, ''Quantum Theory of Atomic Structure'', vol. I, McGraw-Hill, New York (1960), p. 193</ref> that the orbitals arising from the central field, independent-particle model have the following order in increasing energy
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| :<em> 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 5s, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d.</em>
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| Remark: For some atoms the order of 3''d'' and 4''s'' is flipped and for some other atoms the order of 4''d'' and 5''s'' is flipped. Note, for instance, that 4''f'' is not degenerate with 4''d'', which would be the case for hydrogen-like atoms.
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| 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 2''l''+1 orbitals of definite ''l''.
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| For instance, the neon atom (atomic number ''Z'' = 10) has the electronic configuration (''n'' = 1 and ''n'' = 2 shells are completely filled):
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| :<math>{\scriptstyle 1s^2\, 2s^2\, 2p^6,} </math>
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| meaning two electrons are in the 1''s'', two electrons in the 2''s'', and six electrons in the 2''p'' AOs. Similarly, chlorine (''Z'' = 17) has
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| :<math>{\scriptstyle 1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^5\,}. </math>
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| For more details the article [[Periodic Table]] may be consulted.
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| ===Molecular orbitals=== | | ===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.
| | To date the most widely applied method for computing molecular orbitals is a [[Hartree-Fock]] method |
| [[Image:MOdiagram H2.png|right|250px|thumb|MO energy level diagram for H<sub>2</sub>]] | | in which the MO is expanded in [[atomic orbital]]s. Since the term "Hartree-Fock" (HF) is in quantum chemistry almost synonymous with the term "self-consistent field" (SCF), such an MO is often referred to as an SCF-LCAO-MO. |
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| To give the flavor of the theory we look at the simplest molecule: H<sub>2</sub>.
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| The two hydrogen atoms, labeled ''A'' and ''B'', each have one electron (a red arrrow in the figure) in an 1''s'' atomic orbital (AO). These AOs are labeled in the figure ''1s''<sub>A</sub> and ''1s''<sub>B</sub>. When the atoms start to interact the AOs combine linearly to two molecular orbitals:
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| :<math>
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| \sigma_g = N_g (1s_A + 1s_B)\quad \hbox{and}\quad\sigma_u = N_u (1s_A - 1s_B).
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| </math>
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| Earlier in this article a general expression for an LCAO-MO was given. In the present example of the σ<sub>''g''</sub> MO of H<sub>2</sub>, we have ''N''<sub>nuc</sub> = 2, ''n''<sub>''A''</sub> = ''n''<sub>''B''</sub> = 1, χ<sub>''A''1</sub> = 1''s''<sub>''A''</sub>, χ<sub>''B''1</sub> = 1''s''<sub>''B''</sub>, ''c''<sub>''A''1</sub> = ''c''<sub>''B''1</sub> = ''N''<sub>''g''</sub>.
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| 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].
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| Clearly σ<sub>''g''</sub> is symmetric (is even, stays the same) under
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| :<math>
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| \mathrm{{\scriptstyle Reflection:}} \quad 1s_A \longleftrightarrow 1s_B,
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| </math>
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| while σ<sub>''u''</sub> 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).
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| 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.
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| The normalization factors ''N''<sub>''g''</sub> and ''N''<sub>''u''</sub> 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''<sub>''A''1</sub>, ''y''<sub>''A''1</sub>, and ''z''<sub>''A''1</sub> (or over ''x''<sub>''B''1</sub>, ''y''<sub>''B''1</sub>, and ''z''<sub>''B''1</sub> when that is easier).
| | See the section ''computation of molecular orbitals'' in [[molecular orbital]] for a simple example of a case where an MO is almost completely determined by symmetry. In more complicated cases it is necessary to solve the [[Hartree]]-[[Fock]]-[[Roothaan]] equations, which have the form of a generalized [[eigenvalue problem]]. The dimension of this problem is equal to the number of atomic orbitals included in the AO basis (see ''AO basis sets'' in [[atomic orbital]] for more details). |
| :<math>
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| 1=\langle \sigma_g | \sigma_g \rangle = N^2_g\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) =
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| N^2_g(2 + 2 S).
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| </math>
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| We used here that the AOs are normalized to unity and that the overlap integral is real
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| :<math>
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| \langle 1s_A | 1s_B \rangle = \langle 1s_B | 1s_A \rangle \equiv S.
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| </math>
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| Applying the same procedure for ''N''<sub>''u''</sub> we find
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| :<math>
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| N_g = \left[ 2 + 2 S \right]^{-1/2}\quad\hbox{and}\quad N_u = \left[ 2 - 2 S \right]^{-1/2}.
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| </math>
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| At this point one often assumes that ''S'' ≅ 0, so that both normalization factors are equal to ½√2.
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| In order to calculate the energy of an orbital we introduce an effective-one-electron energy operator (Hamiltonian) ''h'',
| | ==History of term orbital== |
| :<math>
| | Orbit is an old noun introduced by [[Johannes Kepler]] in 1609 to describe the trajectories of the earth and the planets. 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 used the word as well, although he was the first to recognize (1913) that an electron orbit is not a trajectory, but a stationary state of the hydrogen atom. |
| \begin{align}
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| E_{\sigma_g} &=
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| \langle \sigma_g |h| \sigma_g \rangle \\
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| & = \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) \\
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| &\equiv \frac{1}{2}(q_A + q_B + \beta_{AB}+ \beta_{BA} ) = q + \beta.
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| \end{align}
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| </math>
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| Since ''A'' and ''B'' are identical atoms equipped with identical AOs, we were allowed to use
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| :<math>
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| q_A = q_B = q = \langle 1s |h| 1s \rangle = E_{1s} \quad\hbox{and}\quad \beta_{AB} = \beta_{BA} = \beta.
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| </math>
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| Likewise
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| :<math>
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| E_{\sigma_u} = \langle \sigma_u |h| \sigma_u \rangle = q - \beta.
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| </math>
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| 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 1''s'' 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.
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| 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<sub>''g''</sub> is a distance |β| below this level, while the energy of σ<sub>''u''</sub> 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 H<sub>2</sub> is 2β (two electrons in the ''bonding MO'' σ<sub>''g''</sub>, ignoring the distance dependence of ''q''). This model predicts that the bonding energy in the one-electron ion H<sub>2</sub><sup>+</sup> is half that of H<sub>2</sub>, which is correct within a 20% margin.
| | After Schrödinger (1926) had solved his wave equation for the hydrogen atom (see [[hydrogen-like atom]]s 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 planetary orbits. As a matter of fact, the angular parts of the hydrogen wave functions are spherical harmonic functions and hence they have the same appearance as these functions. (See [[spherical harmonics]] for a few graphical illustrations). |
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| If we apply the model to He<sub>2</sub>, which has four electrons, we find that we must place two electrons in the bonding orbital and two electrons in the ''antibonding MO'' σ<sub>''u''</sub>, with the total energy being 2β − 2β = 0. So, this simple application of molecular orbital theory predicts that H<sub>2</sub> is bound and that He<sub>2</sub> is not, which is in agreement with the observed facts.
| | 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. |
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| ===Atomic orbital basis sets===
| | In 1932 [[Robert S. Mulliken]] coined the ''noun'' "orbital". Mulliken wrote:<ref>R. S. Mulliken, ''Electronic Structures of Molecules and Valence. II General Considerations'', Physical Review, vol. '''41''', pp. 49-71 (1932) </ref> |
| 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.
| | <blockquote> |
| | ''From here on, one-electron orbital wave functions will be referred to for brevity as orbitals''. |
| | </blockquote> |
| | Note that here, evidently, Mulliken uses "orbital" as relating to "spatial" (non-spin) and defines an orbital as what is now called a "spatial orbital". Then Mulliken went on in the same article to distinguish atomic and molecular orbitals. |
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| A computation of a molecular wave function usually goes through the following steps:
| | Later the somewhat unfortunate term "spinorbital" was introduced for the product functions φ('''r''')α(μ) and φ('''r''')β(μ) in which φ('''r''') has the tautological name "spatial orbital" and α(μ) and β(μ) 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]], juxtaposes ''two-electron spin functions'' and ''two-electron orbital functions''.<ref>W. Heitler, ''Elementary Wave Mechanics'', 2nd edition (1956) Clarendon Press, Oxford, UK.</ref> 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 !) and that an inexperienced reader may easily—and erroneously—interpret the term "two-electron orbital function" as "two-electron orbital". |
| # A geometry of the molecule is chosen (in accordance with the [[Born-Oppenheimer approximation]] the nuclei are clamped in space).
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| # 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 ''n''<sup>6</sup> or ''n''<sup>7</sup>, where ''n'' is the number of AOs.)
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| # An LCAO [[Hartree-Fock]] calculation yields the MO coefficients ''c''<sub>''A i''</sub> and the same number of MOs as AOs (namely ''n'').
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| # The MOs are used in a [[post-Hartree-Fock]] calculation ([[configuration interaction]], [[Møller-Plesset]] perturbation theory, [[coupled cluster]] theory, etc.).
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| | |
| The AOs and MOs spanning the very same orbital subspace of one-electron [[Hilbert space]], <math>{\scriptstyle L^2[\mathbb{R}^3]}</math>, 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.
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| 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 1''s''<sup>2</sup>2''s''<sup>2</sup>2''p'', it cannot be said whether 2''p''<sub>''x''</sub>, 2''p''<sub>''y''</sub>, or 2''p''<sub>''z''</sub> 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 became clear in the late 1960s and early 1970s, search for good AO basis became an important subject of research.
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| In order to explain what a construction of a basis set 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 ("primitive function", "contracted set") 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 [[Gauss type orbital#contracted set|contracted set]] of GTOs. The term "single zeta" is historic and refers back to the days that [[Slater orbital|Slater type orbital]]s 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 givin 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), quadruple zeta (QZ), quintuple zeta (5Z), sextuple zeta (6Z) basis sets all have been proposed and have been constructed.
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| For instance a QZ GTO basis for HCN includes : 4 ''s''-orbitals (contracted sets) for H, 8 ''s''-orbitals for both C and N, 4 ''p''<sub>''x''</sub>-, 4 ''p''<sub>''y''</sub>-, and 4 ''p''<sub>''z''</sub>-orbitals for C and N.
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| In all these cases the construction of the contracted set (AO) involves the determination of the exponential factors of the [[primitive Gaussians]] and the corresponding [[contraction coefficients]].
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| '''(to be continued)'''
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| ==References and notes== | | ==References and notes== |
| <references /> | | <references /> |
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| '''(To be continued)'''
| |
In quantum chemistry, an electron orbital (or more often just orbital) is a synonym for a one-electron function, i.e., a function of a single vector r, the position vector of the electron.[1]
The majority of quantum chemical methods expect that an orbital has a finite norm, i.e., that the orbital is normalizable (quadratically integrable), and hence this requirement is often added to the definition.
In other branches of chemistry, an orbital is often seen as a wave function of an electron, meaning that an orbital is seen as a solution of an (effective) one-electron Schrödinger equation. This point of view is a narrowing of the more general quantum chemical definition, but not contradictory to it. In the past quantum chemists, too, distinguished one-electron functions from orbitals; one-electron functions were fairly arbitrary functions of the position vector r, while orbitals were solutions of certain effective one-electron Schrödinger equations. This distinction faded out in quantum chemistry, resulting in the present definition of "normalizable one-electron function", not necessarily eigenfunctions of a one-electron Hamiltonian.
Usually one distinguishes two kinds of orbitals: atomic orbitals and molecular orbitals.
Atomic orbitals are expressed with respect to one Cartesian system of axes centered on a single atom. Molecular orbitals (MOs) are "spread out" over a molecule. Usually this is a consequence of an MO being a linear combination (weighted sum) of atomic orbitals centered on different atoms.
Definitions of orbitals
Several kinds of orbitals can be distinguished.
Atomic orbital (AO)
See the article atomic orbital
Molecular orbital (MO)
See the article molecular orbital
Spinorbitals
The AOs and MOs defined so far depend only on the spatial coordinate vector r of a single electron. 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 sz of the spin angular momentum operator with eigenvalues ±½.
Spin atomic orbital
The most general spin atomic orbital of an electron is of the form
where r is a vector from the nucleus of the atom to the position of the electron.
In general this function is not an eigenfunction of sz. More common is the use of either
which are eigenfunctions of sz. 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
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:
Chemists express this spin-restriction by stating that two electrons [an electron with spin up (α), and another electron 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).
Applications
Before the advent of electronic computers, orbitals (molecular as well as atomic) were used extensively in qualitative arguments explaining all kinds of 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
Originally AOs were defined as approximate solutions of atomic Schrödinger equations,
see the section solution of the atomic Schrödinger equation in atomic orbital for more on this. Very often atomic orbitals are depicted in polar plots. In fact, the angular parts (functions depending on the spherical polar angles θ and φ) are commonly plotted. The angular parts of atomic orbitals are functions known as spherical harmonics.
As stated, quantum chemists see AOs simply as convenient basis functions for quantum mechanical computations. See the section AO basis sets in the article atomic orbital for more on the use of atomic orbitals in quantum chemistry. See Slater orbital for the explicit analytic form of orbitals and see Gauss type orbitals for a discussion of the type of AOs that are most frequently used in computations.
Molecular orbitals
To date the most widely applied method for computing molecular orbitals is a Hartree-Fock method
in which the MO is expanded in atomic orbitals. Since the term "Hartree-Fock" (HF) is in quantum chemistry almost synonymous with the term "self-consistent field" (SCF), such an MO is often referred to as an SCF-LCAO-MO.
See the section computation of molecular orbitals in molecular orbital for a simple example of a case where an MO is almost completely determined by symmetry. In more complicated cases it is necessary to solve the Hartree-Fock-Roothaan equations, which have the form of a generalized eigenvalue problem. The dimension of this problem is equal to the number of atomic orbitals included in the AO basis (see AO basis sets in atomic orbital for more details).
History of term orbital
Orbit is an old noun introduced by Johannes Kepler in 1609 to describe the trajectories of the earth and the planets. 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 used the word as well, although he was the first to recognize (1913) that an electron orbit is not a trajectory, but a stationary state of the hydrogen atom.
After Schrödinger (1926) had solved his wave equation for the hydrogen atom (see hydrogen-like atoms 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 planetary orbits. As a matter of fact, the angular parts of the hydrogen wave functions are spherical harmonic functions and hence they have the same appearance as these functions. (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". Mulliken wrote:[2]
From here on, one-electron orbital wave functions will be referred to for brevity as orbitals.
Note that here, evidently, Mulliken uses "orbital" as relating to "spatial" (non-spin) and defines an orbital as what is now called a "spatial orbital". Then Mulliken went on in the same article to distinguish atomic and molecular orbitals.
Later the somewhat unfortunate term "spinorbital" was introduced for the product functions φ(r)α(μ) and φ(r)β(μ) in which φ(r) has the tautological name "spatial orbital" and α(μ) and β(μ) 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, juxtaposes two-electron spin functions and two-electron orbital functions.[3] 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 !) and that an inexperienced reader may easily—and erroneously—interpret the term "two-electron orbital function" as "two-electron orbital".
References and notes
- ↑ Here "orbital" is used as a noun. In quantum mechanics, the adjective orbital is often used as a synonym of "spatial" (as in orbital angular momentum), in contrast to spin (as in spin angular momentum).
- ↑ R. S. Mulliken, Electronic Structures of Molecules and Valence. II General Considerations, Physical Review, vol. 41, pp. 49-71 (1932)
- ↑ W. Heitler, Elementary Wave Mechanics, 2nd edition (1956) Clarendon Press, Oxford, UK.
