Thuyet MO

Molecular orbitalJump to: navigation, search In chemistry, a molecular orbital or MO is a region in which an electron may be found in a molecule.[1] Molecular orbitals are described by w

Molecular orbital

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In chemistry, a molecular orbital (or MO) is a region in which an electron may be found

in a molecule.[1] Molecular orbitals are described by wave functions, mathematical

solutions to the Schrödinger wave equation for a molecule, which specify the spatial distribution and energy of up to two electrons within it They can be quantitatively

approximated using the Hartree-Fock or Self-Consistent Field method

Overview

A molecular orbital (MO) specifies 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 (the LCAO-MO method), especially in qualitative or very approximate usage They are invaluable in providing a simple model of bonding in molecules

Most methods in computational chemistry today start 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)

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 In this approach, the molecular orbitals are expressed as linear combinations of atomic orbitals

Molecular orbitals were first introduced by Friedrich Hund and Robert S Mulliken in 1927 and 1928.[2] [3] The linear combination of atomic orbitals approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones.[4] 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

Some properties:

• The number of molecular orbitals is equal to the number the atomic orbitals

included in the linear expansion,

• 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 of the symmetry group,

so the wave functions 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, the symmetry adapted atomic orbitals mix more

if their atomic energy level are closer

[ edit ] Examples

[edit] H2

H2 1sσ* antibonding molecular orbital

As a simple example consider the hydrogen molecule, H2 (see 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:

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 H2 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

[edit] He2

On the other hand, consider the hypothetical molecule of He2 (see molecular orbital

diagram) with the atoms labelled He' and He" Again, the lowest-energy atomic orbitals, 1s' and 1s", do not transform according to the symmetries of the molecule, while the following symmetry adapted atomic orbitals do:

1s' - 1s" Antisymmetric combination: negated by reflection, unchanged by other

operations

1s' + 1s" Symmetric combination: unchanged by all symmetry operations

Similar to the molecule H2, 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 However, in its neutral ground state, each Helium atom contains two electrons in its 1s orbital, combining for a total of four electrons Two electrons fill the lower energy bonding orbital, while the remaining two fill the higher energy antibonding orbital Thus, the resulting electron density around the molecule does not support the formation of a bond between the two atoms (called a sigma bond), and the molecule does therefore not 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

[edit] Noble gases

Considering a hypothetical molecule of He2, since the basis set of atomic orbitals is the same as in the case of H2, 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 H2 + 2 He, so the molecule exists only a short while In general, we find that atoms such as He that have completely full energy shells rarely bond with other atoms

Except for short-lived Van der Waals complexes, there are very few noble gas compounds known

[edit] Ionic bonds

Main article: 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 other's 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 a (mostly) ionic bond

[ edit ] MO diagrams

Main article: Molecular orbital diagram

For more complicated molecules, the wave mechanics approach loses utility in a qualitative understanding of bonding (although is still necessary for a quantitative approach) The qualitative approach of MO uses a molecular orbital diagram 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; have as many unpaired electrons on one energy level as possible before starting to pair them)

[ edit ] More quantitative approach

To obtain quantitative values for the molecular energy levels, one needs to have molecular orbitals which are such that the configuration interaction (CI) expansion converges fast towards the full CI limit The most common method to obtain such functions is the Hartree-Fock method which expresses the molecular orbitals as eigenfunctions of the Hartree-Fock

operator One usually solves this problem by expanding the molecular orbitals as linear combinations of gaussian functions centered on the atomic nuclei (see linear combination

of atomic orbitals and basis set (chemistry)) The equation for the coefficients of these linear combinations is a generalized eigenvalue equation known as the Roothaan equations which are in fact a particular representation of the Hartree-Fock equation

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

MO diagram

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A molecular orbital diagram or MO diagram for short is a qualitative descriptive tool

explaining chemical bonding in molecules in terms of molecular orbital theory in general and the Linear combination of atomic orbitals molecular orbital method (LCAO method) in particular [1] [2] [3] This tool is very well suited for simple diatomic molecules such as dihydrogen, dioxygen and carbon monoxide but becomes more complex when discussing polynuclear molecules such as methane It explains why some molecules exist and not others, how strong bonds are, and what electronic transitions take place

Contents

[hide]

• 1 Dihydrogen MO diagram

• 2 Dihelium MO diagram

• 3 Dilithium MO diagram

• 4 Diboron MO diagram

• 5 Dioxygen MO diagram

• 6 External links

• 7 See also

• 8 References

[ edit ] Dihydrogen MO diagram

The smallest molecule, hydrogen gas exists as dihydrogen (H-H) with a single covalent bond between two hydrogen atoms As each hydrogen atom has a single 1s atomic orbital for its electron, the bond forms by overlap of these two atomic orbitals In figure 1 the two atomic orbitals are depicted on the left and on the right The vertical axis always represents the orbital energies The atomic orbital energy correlates with electronegativity as a more electronegative atom holds an electron more tightly thus lowering its energy MO treatment

is only valid when the atomic orbitals have comparable energy; when they differ greatly the mode of bonding becomes ionic Each orbital is singly occupied with the up and down arrows representing an electron

The two AO's can overlap in two ways depending on their phase relationship The phase of

an orbital is a direct consequence of the wave-like properties of electrons In graphical representations, the orbital phase is depicted either by a plus or minus sign (confusing because there is no relationship to electrical charge) or simply by shading The sign of the phase itself does not have physical meaning except when mixing orbitals to form molecular orbitals

Then two same-sign orbitals have a constructive overlap forming a molecular orbital with the bulk of electron density located between the two nuclei This MO is called the bonding orbital and its energy is lower than that of the original atomic orbitals The orbital is

symmetrical with respect to rotation around the molecular axis (no change) and therefore also called a sigma bond (σ-bond)

The two hydrogen atoms can also interact with each other with their 1s orbitals out-of-phase which leads to destructive cancellation and no electron density between the two nuclei depicted by the so-called nodal plane as the vertical dashed line In this anti-bonding

MO with energy much higher than the original AO's the electrons are located in lobes pointing away from the central axis Like the bonding orbital this orbital is symmetrical but differentiated from it by an asterisk σ* bond

The next step in constructing an MO diagram is filling the molecular orbitals with

electrons With the case of dihydrogen at hand two electrons have to be distributed over a bonding orbital and an anti-bonding orbital Three general rules apply:

• The Aufbau principle states that orbitals are filled starting with the lowest energy

• The Pauli exclusion principle states that the maximum number of electrons

occupying an orbital is two having opposite spins

• Hund's rule states that when there are several MO's with equal energy the electrons fill one MO at a time

Application of these rules for dihydrogen results in having both electrons in the bonding

MO This MO is called the Highest Occupied Molecular Orbital or HOMO which makes the other orbital the Lowest Unoccupied Molecular Orbital or LUMO The electrons in the bond MO are called bonding electrons and any electrons in the antibonding orbital would

be called antibonding electrons The reduction in energy of these electrons is the driving force for chemical bond formation For bonding to exist the bond order defined as:

must have a value larger than 0 The bond order for dihydrogen is (2-0)/2 = 1.

This MO diagram also helps explain how a bond breaks When applying energy to

dihydrogen, a molecular electronic transition takes place when one electron in the bonding

MO is promoted to the antibonding MO The result is that there is no longer a net gain in energy

[ edit ] Dihelium MO diagram

Dihelium (He-He) is a hypothetical molecule and MO theory helps to explain why The

MO diagram for dihelium (2 electrons in each 1s AO) looks very similar to that of

dihydrogen but instead of 2 electrons it is now required to place 4 electrons in the newly formed molecular orbitals

The only way to accomplish this is by occupying the antibonding orbital with two electrons

as well which reduces the bond order ((2-2)/2) to zero and cancels the net energy

stabilization

Another molecule that is precluded based on this principle is diberyllium (beryllium with

electron configuration 1s22s2) On the other hand by removing one electron from dihelium, the stable gas-phase species He2+ ion is formed with bond order 1/2

[ edit ] Dilithium MO diagram

Next up in the periodic table is lithium and MO theory correctly predicts that dilithium is a stable molecule with bond order 1 The 1s MO's are completely filled and do not participate

in bonding

Dilithium is a gas-phase molecule with a much lower bond strength than dihydrogen because the 2s electrons are further removed from the nucleus

[ edit ] Diboron MO diagram

The MO diagram for diboron (B-B electron configuration boron: 1s22s22p1) requires the introduction of an atomic orbital overlap model for p orbitals The three dumbbell-shaped p-orbitals have equal energy and are oriented mutually perpendicular (or orthogonal) The p-orbitals oriented in the x-direction (px) can overlap end-on forming a bonding

(symmetrical) sigma orbital and an antibonding sigma* molecular orbital In contrast to the sigma 1s MO's, The sigma 2p has some non-bonding electron density at either side of the nuclei and the sigma* 2p has some electron density between the nuclei

The other two p-orbitals py and pz can overlap side-on The resulting bonding orbital has its electron density in the shape of two sausages above and below the plane of the molecule The orbital is not symmetrical around the molecular axis and is therefore a pi orbital The antibonding pi orbital (also asymmetrical) has four lobes pointing away from the nuclei Both py and pz orbitals form a pair of pi orbitals equal in energy (degenerate) and can be higher or lower than that of the sigma orbital

In diboron the 1s and 2s electrons do not participate in bonding but the single electrons in the 2p orbitals occupy the 2πpy and the 2πpx MO's resulting in bond order 1 Because the electrons have equal energy (they are degenerate) diboron is a diradical and since the spins are parallel the compound is paramagnetic

Like diboron, dicarbon (C-C electron configuration:1s22s22p2) is a reactive gas-phase molecule Two additional electrons are placed in the 2πp MO's increasing the bond order to

2 The bond order for dinitrogen is three because now two electrons are added in the 2σ

MO as well

[ edit ] Dioxygen MO diagram

MO treatment of dioxygen is different from that of the previous diatomic molecules

because the pσ MO is now lower in energy than the 2π orbitals This is attributed to

interaction between the 2s MO and the 2pz MO [4] Distributing 8 electrons over 6

molecular orbitals leaves the final two electrons as a degenerate pair in the 2pπ*

antibonding orbitals resulting in a bond order of 2 Just as diboron, this type of dioxygen called triplet oxygen is a paramagnetic diradical When both HOMO electrons pair up the other oxygen type is called singlet oxygen

The bond order decreases and the bond length increases in the order O2+ (112.2 pm), O2

(121 pm), O2- (128 pm) and O22- (149 pm) [4]

In difluorine two additional electrons occupy the 2pπ* with a bond order of 1 In dineon

Ne2 (as with dihelium) the number of bonding electrons equals the number of antibonding electrons and this compound does not exist

Linear combination of atomic orbitals

molecular orbital method

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(Redirected from Linear combination of atomic orbitals)

Jump to: navigation, search

A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic

orbitals and a technique for calculating molecular orbitals in quantum chemistry [1] In quantum mechanics, electron configurations of atoms are described as wavefunctions In mathematical sense, these wave functions are the basis set of functions, the basis functions, which describe the electrons of a given atom In chemical reactions, orbital wavefunctions are modified, i.e the electron cloud shape is changed, according to the type of atoms participating in the chemical bond