|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{see also|Molecular orbital theory}}
| | The Syracuse Christian Living Examiner's site is really a place where users, devotions, events, reviews, and ideas are distributed. So just why not discuss that which you've got to say?<br><br> |
| [[File:Orbitals acetylene.jpg|right|thumb|Complete [[acetylene]] (H–C≡C–H) molecular orbital set. The left column shows MO's which are occupied in the ground state, with the lowest-energy orbital at the top. The white and grey line visible in some MO's is the molecular axis passing through the nuclei. The orbital wave functions are positive in the red regions and negative in the blue. The right column shows virtual MO's which are empty in the ground state, but may be occupied in excited states.]]
| |
|
| |
|
| In [[chemistry]], a '''molecular orbital''' (or '''MO''') is a [[Function (mathematics)|mathematical function]] describing the wave-like behavior of an [[electron]] in a [[molecule]]. This function can be used to calculate chemical and physical properties such as the probability of finding an [[electron]] in any specific region. The term ''orbital'' was introduced by [[Robert S. Mulliken]] in 1932 as an abbreviation for ''one-electron orbital wave function''.<ref>{{cite journal
| | Free [http://www.advertisingservicesguide.com/ Advertising Agencies] websites provide you with an opportunity to take a look at many advertising styles of written ads. You view a lot of different powerful advertisements promoting the same product but differently! A great headline should make the viewer click it!<br><br><br><br>Larry Page was contained in a conference call to discuss its third quarter profits which was leaked accidentally in front of plan on Thursday (October 18, 2012). The President of Google found the quarter strong, noting that sales rose 45% on the year.<br><br>Packages like google adsense enable you to generate by either clicks or page-thoughts. Ticks make reference to the clicks of the visitor of the website for the ads you have inserted. The percent of ticks per impact depends upon the click-through-rate or CTR. You are paid per-click towards the ads in your corner. The quantity varies is dependent upon lots of variables. Which means it is never constant.<br><br>Now once you found the correct destination for a advertise your organization or product, then its time up with a very excellent Ad. Remember when creating your little Advertisement, you're selling the click for the Advertisement, you need people to click in your Offer because of the rewards to your prospects. While writing an Ad make an effort to list the advantages and not really much the features.<br><br>"We're happy we've had more allies contribute more income than any strategy beforehand in Tennessee at this stage, and we assume that's a testament to the broad, strong assistance Statement has throughout the state, and that's barely purchasing the election," Ingram said.<br><br>Start of by writing some headlines that clearly speak your offer, reward or USP. Then try to find ways to make them a lot more effective with the addition of an electrical word or two and checking when they include at least 3 of the rules presented above. |
| | last=Mulliken | first=Robert S.
| |
| | title=Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations
| |
| |date=July 1932
| |
| | journal=[[Physical Review]]
| |
| | volume=41 | issue=1 | pages=49–71
| |
| | bibcode = 1932PhRv...41...49M
| |
| | doi = 10.1103/PhysRev.41.49
| |
| }}</ref> At an elementary level it is used to describe the ''region'' of space in which the function has a significant amplitude. Molecular orbitals are usually constructed by combining [[atomic orbital]]s or [[hybrid orbital]]s from each [[atom]] of the molecule, or other molecular orbitals from groups of atoms. They can be quantitatively calculated using the [[Hartree–Fock method|Hartree–Fock]] or self-consistent field (SCF) methods.
| |
| | |
| ==Overview==
| |
| A molecular orbital (MO) can be used to represent the regions in a [[molecule]] where an [[electron]] occupying that orbital is likely to be found. Molecular orbitals are obtained from the combination of atomic orbitals, which predict the location of an electron in an atom. A molecular orbital can specify the [[electron configuration]] of a molecule: the spatial distribution and energy of one (or one pair of) electron(s). Most commonly an MO is represented as a [[Linear combination of atomic orbitals molecular orbital method|linear combination of atomic orbitals]] (the LCAO-MO method), especially in qualitative or very approximate usage. They are invaluable in providing a simple model of bonding in molecules, understood through [[molecular orbital theory]].
| |
| Most present-day methods in [[computational chemistry]] begin by calculating the MOs of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. In the case of two electrons occupying the same orbital, the [[Pauli principle]] demands that they have opposite spin. Necessarily this is an approximation, and highly accurate descriptions of the molecular electronic wave function do not have orbitals (see [[configuration interaction]]).
| |
| | |
| ==Formation of molecular orbitals==
| |
| Molecular orbitals arise from allowed interactions between [[atomic orbital]]s, which are allowed if the symmetries (determined from [[group theory]]) of the atomic orbitals are compatible with each other. Efficiency of atomic orbital interactions is determined from the [[Orbital overlap|overlap]] (a measure of how well two orbitals constructively interact with one another) between two atomic orbitals, which is significant if the atomic orbitals are close in energy. Finally, the number of molecular orbitals that form must equal the number of atomic orbitals in the atoms being combined to form the molecule.
| |
| | |
| ==Qualitative discussion==
| |
| For an imprecise, but qualitatively useful, discussion of the molecular structure, the molecular orbitals can be obtained from the "[[Linear combination of atomic orbitals molecular orbital method]]" [[ansatz]]. Here, the molecular orbitals are expressed as [[linear combination]]s of [[atomic orbital]]s.
| |
| | |
| ===Linear combinations of atomic orbitals (LCAO)===
| |
| | |
| {{main|Linear combination of atomic orbitals}}
| |
| Molecular orbitals were first introduced by [[Friedrich Hund]]<ref>F. Hund, "Zur Deutung einiger Erscheinungen in den Molekelspektren" [On the interpretation of some phenomena in molecular spectra] ''Zeitschrift für Physik'', vol. 36, pages 657-674 (1926).</ref><ref>F. Hund, "Zur Deutung der Molekelspektren," ''Zeitschrift für Physik'', Part I, vol. 40, pages 742-764 (1927); Part II, vol. 42, pages 93–120 (1927); Part III, vol. 43, pages 805-826 (1927); Part IV, vol. 51, pages 759-795 (1928); Part V, vol. 63, pages 719-751 (1930).</ref> and [[Robert S. Mulliken]]<ref>R. S. Mulliken, "Electronic states. IV. Hund's theory; second positive nitrogen and Swan bands; alternate intensities," ''Physical Review'', vol. 29, pages 637 - 649 (1927).</ref><ref>R. S. Mulliken, "The assignment of quantum numbers for electrons in molecules," ''Physical Review'', vol. 32, pages 186 - 222 (1928).</ref> in 1927 and 1928.<ref>[[Friedrich Hund]] and Chemistry, [[Werner Kutzelnigg]], on the occasion of Hund's 100th birthday, [[Angewandte Chemie International Edition]], 35, 573 - 586, (1996)</ref><ref>[[Robert S. Mulliken]]'s Nobel Lecture, [[Science (journal)|Science]], 157, no. 3785, 13 - 24, (1967). Available on-line at: [http://nobelprize.org/nobel_prizes/chemistry/laureates/1966/mulliken-lecture.pdf Nobelprize.org .]</ref> The [[linear combination of atomic orbitals]] or "LCAO" approximation for molecular orbitals was introduced in 1929 by [[John Lennard-Jones|Sir John Lennard-Jones]].<ref>[[John Lennard-Jones|Sir John Lennard-Jones]], "The electronic structure of some diatomic molecules," ''[[Transactions of the Faraday Society]]'', vol. 25, pages 668-686 (1929).</ref> His ground-breaking paper showed how to derive the electronic structure of the [[fluorine]] and [[oxygen]] molecules from quantum principles. This qualitative approach to molecular orbital theory is part of the start of modern [[quantum chemistry]].
| |
| Linear combinations of atomic orbitals (LCAO) can be used to estimate the molecular orbitals that are formed upon bonding between the molecule’s constituent atoms. Similar to an atomic orbital, a Schrödinger equation, which describes the behavior of an electron, can be constructed for a molecular orbital as well. Linear combinations of atomic orbitals, or the sums and differences of the atomic wavefunctions, provide approximate solutions to the [[Hartree–Fock method|Hartree-Fock equations]] which correspond to the independent-particle approximation of the molecular [[Schrödinger equation]]. For simple diatomic molecules, the wavefunctions obtained are represented mathematically by the equations
| |
| | |
| :<math>\Psi = c_a \psi_a + c_b \psi_b</math>
| |
| :<math>\Psi^* = c_a \psi_a - c_b \psi_b</math>
| |
| | |
| where <math>\Psi</math> and <math>\Psi^*</math> are the molecular wavefunctions for the bonding and antibonding molecular orbitals, respectively, <math>\psi_a</math> and <math>\psi_b</math> are the atomic wavefunctions from atoms a and b, respectively, and <math>c_a</math> and <math>c_b</math> are adjustable coefficients. These coefficients can be positive or negative, depending on the energies and symmetries of the individual atomic orbitals. As the two atoms become closer together, their atomic orbitals overlap to produce areas of high electron density, and, as a consequence, molecular orbitals are formed between the two atoms. The atoms are held together by the electrostatic attraction between the positively charged nuclei and the negatively charged electrons occupying bonding molecular orbitals.<ref>Gary L. Miessler; Donald A. Tarr. Inorganic Chemistry. Pearson Prentice Hall, 3rd ed., 2004.</ref>
| |
| | |
| ===Bonding, antibonding, and nonbonding MOs===
| |
| When atomic orbitals interact, the resulting molecular orbital can be of three types: bonding, antibonding, or nonbonding.
| |
| | |
| Bonding MOs:
| |
| | |
| *Bonding interactions between atomic orbitals are constructive (in-phase) interactions.
| |
| *Bonding MOs are lower in energy than the atomic orbitals that combine to produce them.
| |
| Antibonding MOs:
| |
| *Antibonding interactions between atomic orbitals are destructive (out-of-phase) interactions, with a [[Node (physics)|nodal plane]] where the wavefunction of the antibonding orbital is zero between the two interacting atoms
| |
| *Antibonding MOs are higher in energy than the atomic orbitals that combine to produce them.
| |
| Nonbonding MOs:
| |
| *Nonbonding MOs are the result of no interaction between atomic orbitals because of lack of compatible symmetries.
| |
| *Nonbonding MOs will have the same energy as the atomic orbitals of one of the atoms in the molecule.
| |
| | |
| ===Sigma and pi labels for MOs===<!-- this section-title is linked from some redirect pages...do not change it here without also updating them -->
| |
| The type of interaction between atomic orbitals can be further categorized by the molecular-orbital symmetry labels σ (sigma), π (pi), δ (delta), φ (phi), γ (gamma) etc. paralleling the symmetry of the atomic orbitals s, p, d, f and g. The number of nodal planes containing the internuclear axis between the atoms concerned is zero for σ MOs, one for π, two for δ, etc. | |
| | |
| ====σ Symmetry====
| |
| {{details|Sigma bond}}
| |
| A MO with σ symmetry results from the interaction of either two atomic s-orbitals or two atomic p<sub>z</sub>-orbitals. An MO will have σ-symmetry if the orbital is symmetrical with respect to the axis joining the two nuclear centers, the internuclear axis. This means that rotation of the MO about the internuclear axis does not result in a phase change. A σ* orbital, sigma antibonding orbital, also maintains the same phase when rotated about the internuclear axis. The σ* orbital has a nodal plane that is between the nuclei and perpendicular to the internuclear axis.<ref>Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, p. 29-33.</ref>
| |
| | |
| ====π Symmetry====
| |
| {{details|Pi bond}}
| |
| A MO with π symmetry results from the interaction of either two atomic p<sub>x</sub> orbitals or p<sub>y</sub> orbitals. An MO will have π symmetry if the orbital is asymmetrical with respect to rotation about the internuclear axis. This means that rotation of the MO about the internuclear axis will result in a phase change. There is one nodal plane containing the internuclear axis, if [[Atomic orbital#Real orbitals|real orbitals]] are considered.
| |
| | |
| A π* orbital, pi antibonding orbital, will also produce a phase change when rotated about the internuclear axis. The π* orbital also has a second nodal plane between the nuclei.<ref>Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, p. 29-33.</ref><ref>Peter Atkins; Julio De Paula. Atkins’ Physical Chemistry. Oxford University Press, 8th ed., 2006.</ref><ref>Yves Jean; Francois Volatron. An Introduction to Molecular Orbitals. Oxford University Press, 1993.</ref><ref>Michael Munowitz, Principles of Chemistry, Norton & Company, 2000, p. 229-233.</ref>
| |
| | |
| ====δ Symmetry====
| |
| {{details|Delta bond}}
| |
| A MO with δ symmetry results from the interaction of two atomic d<sub>xy</sub> or d<sub>x<sup>2</sup>-y<sup>2</sub></sub> orbitals. Because these molecular orbitals involve low-energy d atomic orbitals, they are seen in [[transition metal|transition-metal]] complexes. A δ bonding orbital has two nodal planes containing the internuclear axis, and a δ* antibonding orbital also has a third nodal plane between the nuclei.
| |
| | |
| ====φ Symmetry====<!-- this section-title is linked from some redirect pages...do not change it here without also updating them -->
| |
| {{multiple image
| |
| | footer = Suitably aligned f atomic orbitals overlap to form phi molecular orbital (a phi bond)
| |
| | width = 120
| |
| | image1 = Phi-bond-f-orbitals-2D.png
| |
| | image2 = Phi-bond-boundary-surface-diagram-2D.png
| |
| | caption = Suitably aligned f atomic orbitals can overlap to form a phi molecular orbital (a phi bond)
| |
| }}
| |
| Theoretical chemists have conjectured that higher-order bonds, such as phi bonds corresponding to overlap of f atomic orbitals, are possible. There is as of 2005 only one known example of a molecule purported to contain a phi bond (a U−U bond, in the molecule [[diuranium|U<sub>2</sub>]]).<ref>{{cite journal|last=Gagliardi|first=Laura|coauthors=Roos, Björn O.|title=Quantum chemical calculations show that the uranium molecule U2 has a quintuple bond|journal=Nature|year=2005|volume=433|pages=848–851|doi=10.1038/nature03249|url=http://archive-ouverte.unige.ch/unige:3652|bibcode = 2005Natur.433..848G }}</ref>
| |
| | |
| ===Gerade and ungerade symmetry===
| |
| For molecules that possess a center of inversion ([[centrosymmetry|centrosymmetric molecules]]) there are additional labels of symmetry that can be applied to molecular orbitals.
| |
| Centrosymmetric molecules include:
| |
| *[[Homonuclear molecule|Homonuclear]] diatomics, X<sub>2</sub>
| |
| *[[Octahedral molecular geometry|Octahedral]], EX<sub>6</sub>
| |
| *[[square planar molecular geometry|Square planar]], EX<sub>4</sub>.
| |
| Non-centrosymmetric molecules include:
| |
| * [[Heteronuclear molecule|Heteronuclear]] diatomics, XY
| |
| *[[Tetrahedral molecular geometry|Tetrahedral]], EX<sub>4</sub>.
| |
| If inversion through the center of symmetry in a molecule results in the same phases for the molecular orbital, then the MO is said to have gerade (g) symmetry, from the German word for even.
| |
| If inversion through the center of symmetry in a molecule results in a phase change for the molecular orbital, then the MO is said to have ungerade (u) symmetry, from the German word for odd.
| |
| For a bonding MO with σ-symmetry, the orbital is σ<sub>g</sub> (s' + s<nowiki>''</nowiki> is symmetric), while an antibonding MO with σ-symmetry the orbital is σ<sub>u</sub>, because inversion of s' – s<nowiki>''</nowiki> is antisymmetric.
| |
| For a bonding MO with π-symmetry the orbital is π<sub>u</sub> because inversion through the center of symmetry for would produce a sign change (the two p atomic orbitals are in phase with each other but the two lobes have opposite signs), while an antibonding MO with π-symmetry is π<sub>g</sub> because inversion through the center of symmetry for would not produce a sign change (the two p orbitals are antisymmetric by phase).<ref>Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, p. 29-33.</ref>
| |
| | |
| ===MO diagrams===
| |
| {{main|Molecular orbital diagram}}
| |
| | |
| The qualitative approach of MO analysis uses a molecular orbital diagram to visualize bonding interactions in a molecule. In this type of diagram, the molecular orbitals are represented by horizontal lines; the higher a line the higher the energy of the orbital, and degenerate orbitals are placed on the same level with a space between them. Then, the electrons to be placed in the molecular orbitals are slotted in one by one, keeping in mind the Pauli exclusion principle and Hund's rule of maximum multiplicity (only 2 electrons, having opposite spins, per orbital; place as many unpaired electrons on one energy level as possible before starting to pair them). For more complicated molecules, the wave mechanics approach loses utility in a qualitative understanding of bonding (although is still necessary for a quantitative approach).
| |
| Some properties:
| |
| * A basis set of orbitals includes those atomic orbitals that are available for molecular orbital interactions, which may be bonding or antibonding
| |
| * The number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion or the basis set
| |
| * If the molecule has some symmetry, the degenerate atomic orbitals (with the same atomic energy) are grouped in linear combinations (called '''symmetry-adapted atomic orbitals (SO)'''), which belong to the [[Representation theory of finite groups|representation]] of the [[symmetry group]], so the [[wave function]]s that describe the group are known as '''symmetry-adapted linear combinations''' ('''SALC''').
| |
| * The number of molecular orbitals belonging to one group representation is equal to the number of symmetry-adapted atomic orbitals belonging to this representation
| |
| * Within a particular [[Representation theory of finite groups|representation]], the symmetry-adapted atomic orbitals mix more if their atomic [[energy level]]s are closer.
| |
| | |
| ===Bonding in molecular orbitals===
| |
| | |
| ====Orbital degeneracy====
| |
| {{main|Degenerate orbital}}
| |
| Molecular orbitals are said to be degenerate if they have the same energy. For example, in the homonuclear diatomic molecules of the first ten elements, the molecular orbitals derived from the p<sub>x</sub> and the p<sub>y</sub> atomic orbitals result in two degenerate bonding orbitals (of low energy) and two degenerate antibonding orbitals (of high energy).<ref>Gary L. Miessler; Donald A. Tarr. Inorganic Chemistry. Pearson Prentice Hall, 3rd ed., 2004.</ref>
| |
| | |
| ====Ionic bonds====
| |
| {{main|Ionic bond}}
| |
| | |
| When the energy difference between the atomic orbitals of two atoms is quite large, one atom's orbitals contribute almost entirely to the bonding orbitals, and the others atom’s orbitals contribute almost entirely to the antibonding orbitals. Thus, the situation is effectively that some electrons have been transferred from one atom to the other. This is called an (mostly) [[ionic bond]].
| |
| | |
| ====Bond order ====
| |
| {{main|Bond order}}
| |
| The bond order, or number of bonds, of a molecule can be determined by combining the number of electrons in bonding and antibonding molecular orbitals. A pair of electrons in a bonding orbital creates a bond, whereas a pair of electrons in an antibonding orbital negates a bond. For example, N<sub>2</sub>, with eight electrons in bonding orbitals and two electrons in antibonding orbitals, has a bond order of three, which constitutes a triple bond.
| |
| | |
| [[Bond strength]] is proportional to bond order—a greater amount of bonding produces a more stable bond—and [[bond length]] is inversely proportional to it—a stronger bond is shorter.
| |
| | |
| There are rare exceptions to the requirement of molecule having a positive bond order. Although Be<sub>2</sub> has a bond order of 0 according to MO analysis, there is experimental evidence of a highly unstable Be<sub>2</sub> molecule having a bond length of 245 pm and bond energy of 10 kJ/mol.<ref>Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, p. 29-33.</ref>
| |
| | |
| ====HOMO and LUMO====
| |
| {{main|HOMO/LUMO}}
| |
| The highest occupied molecular orbital and lowest unoccupied molecular orbital are often referred to as the HOMO and LUMO, respectively. The difference of the energies of the HOMO and LUMO, termed the [[band gap]], can sometimes serve as a measure of the [[Excited state|excitability]] of the molecule: The smaller the energy the more easily it will be excited.
| |
| | |
| ==Molecular orbital examples==
| |
| | |
| ===Homonuclear diatomics===
| |
| Homonuclear diatomic MOs contain equal contributions from each atomic orbital in the basis set. This is shown in the homonuclear diatomic MO diagrams for H<sub>2</sub>, He<sub>2</sub>, and Li<sub>2</sub>, all of which containing symmetric orbitals.<ref>Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, p. 29-33.</ref>
| |
| | |
| ====H<sub>2</sub>====
| |
| [[File:H2OrbitalsAnimation.gif|thumb|right|300px|Electron [[wavefunction]]s for the [[atomic orbital|1s orbital]] of a lone hydrogen atom (left and right) and the corresponding bonding (bottom) and antibonding (top) molecular orbitals of the H<sub>2</sub> molecule. The [[real part]] of the wavefunction is the blue curve, and the [[imaginary part]] is the red curve. The red dots mark the locations of the nuclei. The electron wavefunction oscillates according to the [[Schrödinger wave equation]], and orbitals are its [[standing wave]]s. The standing wave frequency is proportional to the orbital's kinetic energy. (This plot is a one-dimensional slice through the three-dimensional system.)]]
| |
| As a simple MO example consider the [[hydrogen]] molecule, H<sub>2</sub> (see [[MO diagram#Diatomic MO diagrams|molecular orbital diagram]]), with the two atoms labelled H' and H". The lowest-energy atomic orbitals, 1s' and 1s", do not transform according to the symmetries of the molecule. However, the following symmetry adapted atomic orbitals do:
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !1s' – 1s"
| |
| |Antisymmetric combination: negated by reflection, unchanged by other operations
| |
| |-
| |
| !1s' + 1s"
| |
| |Symmetric combination: unchanged by all symmetry operations
| |
| |}
| |
| | |
| The symmetric combination (called a bonding orbital) is lower in energy than the basis orbitals, and the antisymmetric combination (called an [[antibonding]] orbital) is higher. Because the H<sub>2</sub> molecule has two electrons, they can both go in the bonding orbital, making the system lower in energy (and, hence, more stable) than two free hydrogen atoms. This is called a [[covalent bond]]. The [[bond order]] is equal to the number of bonding electrons minus the number of antibonding electrons, divided by 2. In this example, there are 2 electrons in the bonding orbital and none in the antibonding orbital; the bond order is 1, and there is a single bond between the two hydrogen atoms.
| |
| | |
| ====He<sub>2</sub>====
| |
| On the other hand, consider the hypothetical molecule of He<sub>2</sub> with the atoms labeled He' and He". As with H<sub>2</sub>, the lowest-energy atomic orbitals are the 1s' and 1s", and do not transform according to the symmetries of the molecule, while the symmetry adapted atomic orbitals do. The symmetric combination—the bonding orbital—is lower in energy than the basis orbitals, and the antisymmetric combination—the antibonding orbital—is higher. Unlike H<sub>2</sub>, with two valence electrons, He<sub>2</sub> has four in its neutral ground state. Two electrons fill the lower-energy bonding orbital, σ<sub>g</sub>(1s), while the remaining two fill the higher-energy antibonding orbital, σ<sub>u</sub>*(1s). Thus, the resulting electron density around the molecule does not support the formation of a bond between the two atoms; without a stable bond holding the atoms together, molecule would not be expected to exist. Another way of looking at it is that there are two bonding electrons and two antibonding electrons; therefore, the bond order is 0 and no bond exists (the molecule has one bound state supported by the Van der Waals potential).{{cn|date=January 2014}}
| |
| | |
| ====Li<sub>2</sub>====
| |
| [[Dilithium]] Li<sub>2</sub> is formed from the overlap of the 1s and 2s atomic orbitals (the basis set) of two Li atoms. Each Li atom contributes three electrons for bonding interactions, and the six electrons fill the three MOs of lowest energy, σ<sub>g</sub>(1s), σ<sub>u</sub>*(1s), and σ<sub>g</sub>(2s). Using the equation for bond order, it is found that dilithium has a bond order of one, a single bond.
| |
| | |
| ====Noble gases====
| |
| Considering a hypothetical molecule of He<sub>2</sub>, since the basis set of atomic orbitals is the same as in the case of H<sub>2</sub>, we find that both the bonding and antibonding orbitals are filled, so there is no energy advantage to the pair. HeH would have a slight energy advantage, but not as much as H<sub>2</sub> + 2 He, so the molecule is very unstable and exists only briefly before decomposing into hydrogen and helium. In general, we find that atoms such as He that have full energy shells rarely bond with other atoms. Except for short-lived [[Van der Waals bonding|Van der Waals complexes]], there are very few [[noble gas compound]]s known.
| |
| | |
| ===Heteronuclear diatomics===
| |
| While MOs for homonuclear diatomic molecules contain equal contributions from each interacting atomic orbital, MOs for heteronuclear diatomics contain different atomic orbital contributions. Orbital interactions to produce bonding or antibonding orbitals in heteronuclear diatomics occur if there is sufficient overlap between atomic orbitals as determined by their symmetries and similarity in orbital energies.
| |
| | |
| ====HF====
| |
| In [[hydrogen fluoride]] HF overlap between the H 1s and F 2s orbitals is allowed by symmetry but the difference in energy between the two atomic orbitals prevents them from interacting to create a molecular orbital. Overlap between the H 1s and F 2p<sub>z</sub> orbitals is also symmetry allowed, and these two atomic orbitals have a small energy separation. Thus, they interact, leading to creation of σ and σ* MOs and a molecule with a bond order of 1. Since HF is a non-centrosymmetric molecule, the symmetry labels g and u do not apply to its molecular orbitals.<ref>Catherine E. Housecroft, Alan G, Sharpe, Inorganic Chemistry, Pearson Prentice Hall; 2nd Edition, 2005, p. 41-43.</ref>
| |
| | |
| ==Quantitative approach==
| |
| To obtain quantitative values for the molecular [[energy level]]s, one needs to have molecular orbitals that are such that the [[configuration interaction]] (CI) expansion converges fast towards the [[Full configuration interaction|full CI]] limit. The most common method to obtain such functions is the [[Hartree–Fock method]], which expresses the molecular orbitals as [[eigenfunction]]s of the [[Fock operator]]. One usually solves this problem by expanding the molecular orbitals as linear combinations of [[Gaussian function]]s centered on the [[atomic nucleus|atomic nuclei]] (see [[Linear combination of atomic orbitals molecular orbital method|linear combination of atomic orbitals]] and [[basis set (chemistry)]]). The equation for the coefficients of these [[linear combination]]s is a generalized [[eigenvalue]] equation known as the [[Roothaan equations]], which are in fact a particular representation of the Hartree-Fock equation. There are a number of programs in which quantum chemical calculations of MOs can be performed, including [[Spartan (software)|Spartan]] and HyperChem.
| |
| | |
| Simple accounts often suggest that experimental molecular orbital energies can be obtained by the methods of [[ultra-violet photoelectron spectroscopy]] for valence orbitals and [[X-ray photoelectron spectroscopy]] for core orbitals. This, however, is incorrect as these experiments measure the ionization energy, the difference in energy between the molecule and one of the ions resulting from the removal of one electron. Ionization energies are linked approximately to orbital energies by [[Koopmans' theorem]]. While the agreement between these two values can be close for some molecules, it can be very poor in other cases.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| ==External links==
| |
| *[http://www.falstad.com/qmmo/ Java molecular orbital viewer] shows orbitals of hydrogen molecular ion.
| |
| *[http://winter.group.shef.ac.uk/orbitron/ The orbitron], a visualization of all atomic, and some molecular and hybrid orbitals
| |
| *[http://sourceforge.net/projects/xeo/ xeo] Visualizations of some atomic and molecular atoms
| |
| *[http://www.webreader.net/animations.htm Simulations of molecules with electrons caught in molecular orbital] (Simulations run on PC only.)
| |
| *[http://www.lct.jussieu.fr/pagesperso/orbimol OrbiMol] Molecular orbital database.
| |
| | |
| {{Chemical bonds}}
| |
| | |
| {{DEFAULTSORT:Molecular Orbital}}
| |
| [[Category:Molecular physics]]
| |
| [[Category:Quantum chemistry]]
| |
| [[Category:Theoretical chemistry]]
| |
| [[Category:Computational chemistry]]
| |
| [[Category:Chemical bonding]]
| |