Molecular orbital theoryMolecular orbital method(Molecular Orbital Theory) or MO method, proposed by American chemist R.S. Mulliken and German physicist F. Hund in 1932, is a method to describe the state of electrons in polyatomic molecules.Molecular orbital theory is modernCovalent Bond Theory One of its key points is:moleculeTo discuss the structure of molecules. It is believed that after atoms form molecules, electrons no longer belong to individualsAtomic orbitalIt belongs to the whole moleculemolecular orbital, the molecular orbital is polycentric;Molecular orbitals are composed of atomic orbitals. The formation of molecular orbitals follows the principles of energy approximation, symmetry consistency (matching), and maximum overlap, which are commonly referred to as the "three principles of bonding";The principle of filling molecular orbitals with electrons in molecules also obeysPrinciple of minimum energy、Pauli exclusion principleandHunt rule。[1]
In 1927Valence bond theoryIt was proposed, and then, influenced by Hunt, Maliken, Slade and John Lennard Jones, molecular orbitals began to be generated.Therefore, at the beginning, the molecular orbital theory was called the Hunt Maliken theory.The concept of "track" was first proposed by Mariken in 1932.
By 1933, the molecular orbital theory had been widely accepted, and was considered to be an effective and useful theory.In fact, according to German physical chemistsHuckel The first document using molecular orbital theory was published by Leonard Jones in 1929.The first quantitative calculation literature using molecular orbital theory was published by Coulson in 1938 to solve the electronic wave function of hydrogen molecule using self consistent field theory.
By 1950, molecular orbitals were completely defined asEigenfunctionThis is the sign that the molecular orbital theory has developed into a rigorous scientific theory.The Hartree Fock method is a more rigorous treatment of molecular orbital theory,HF method is a method used to calculate the electronic structure of atoms. However, in molecular computing, molecular orbits are expanded according to a set of basis sets of atomic orbits, and Rothan equation is developed. On this basis, various ab initio quantum chemical calculation methods have been developed.At the same time, molecular orbital theory has also been applied to a semi empirical calculation that uses more approximate methods, called semi empirical quantum chemical calculation method.[2]
Introduction to Theory
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one kindChemical bondTheory is the natural extension of atomic orbital theory to molecules.Its basic view is that there is a single electron's own behavior in physics, which is only affected by the average field of the atomic nucleus and other electrons in the molecule, andPauli exclusion principleRestriction of;Mathematically, it attempts to simplify the unsolvable equation of motion of multiple electrons into a single electron equation.Therefore, molecular orbital theory is based onOne electron approximationBasedChemical bond theory。Describing the behavior of a single electronwave functionOrbit (or orbital function), corresponding single electron energy scaleenergy level。For any molecule, if a series of molecular orbitals and energy levels are obtained, the molecular structure can be discussed like the atomic structure, and related to the molecular propertiesSystematic explanation。Sometimes, even based on the partial approximate molecular orbitals and energy levels obtained from the rough calculation scheme, useful qualitative results can be analyzed.
Track Introduction
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1. When atoms form molecules, all electrons contribute. Electrons in molecules no longer belong to an atom, but move in the whole molecular space.Space of electrons in moleculesmotion state The corresponding molecular orbital can be usedwave functionψ (called molecular orbital).The main differences between molecular orbitals and atomic orbitals are:
(1) In atoms, the motion of electrons is affected only by one nucleus,Atomic orbitalIt is a single core system;In molecules, electrons move under the action of all nuclear potential fields, and molecular orbits are multi-core systems.
Molecular orbital theory
(2) The names of atomic orbitals are represented by s, p, d... symbols, while the names of molecular orbitals are represented by σ, π, δ... symbols accordingly.
2. Molecular orbitals can be determined byOrbital wave functionLinear combination of atomic orbitals (LCAO).Several atomic orbitals can be combined into several molecular orbitals. Some of the molecular orbitals are formed by superposition of two symmetrically matched atomic orbitals. The probability density of electrons between the two nuclei increases, and their energy is lower than the original atomic orbital energy, which is conducive to bonding. They are called bonding molecular orbitals, such as σ, π orbitals (axisymmetric orbitals)；At the same time, the two atomic orbitals matched by these symmetries will also be subtracted to form another kind of molecular orbital. The result is that the probability density of electrons between the two nuclei is very small, and its energy is higher than the original atomic orbital energy, which is not conducive to bonding. It is called anti bonding molecular orbital, such as σ *, π * orbitals（Mirror symmetryOrbits, anti bond orbits are often marked with "*" to matchBonding orbitalDifference).Another special case is that the molecular orbitalAtomic orbitalThe spatial symmetry of is not matched, the atomic orbitals do not overlap effectively, and the energy of the combined molecular orbitals is not significantly different from that of the atomic orbitals before combination. The obtained molecular orbitals are called non bonded molecular orbitals.
3. The arrangement of electrons in molecular orbits also follows the same principle as that of atomic orbits, namely Pauli exclusion principlePrinciple of minimum energyAnd Hund rules.For specific arrangement, the molecular orbitalenergy levelSequence.The current order mainly depends onMolecular spectrumExperiment to determine.
Linear combination principle
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The principle of energy approximation, the principle of symmetry matching and the principle of maximum overlap of atomic orbitals when combining atomic orbitals to form molecular orbitals are called the three principles of bonding.[1]
Symmetry matching principle
Only symmetrically matched atomic orbitals can be combined into molecular orbitals, which is called symmetry matching principle.There are various types of atomic orbitals such as s, p, d, etcGeometryIt can be seen that they have different spatial symmetry for some points, lines, planes, etc.Whether the symmetry matches can be determined according to the symmetry of the positive and negative signs of the lobe in the angular distribution diagram of the two atomic orbits with respect to the bond axis (set as the x-axis) or a plane containing the bond axis.
Energy approximation principle
In symmetry matched atomic orbitals, only atomic orbitals with similar energies can be combined into effective molecular orbitals, and the closer the energies are, the better. This is called the energy approximation principle.
Principle of maximum overlap of tracks
Two Symmetrical MatchesAtomic orbitalIn linear combination, the greater the overlap, the lower the energy of the combined molecular orbital, and the stronger the chemical bond formed, which is called the maximum overlap principle of orbital.Among the three principles mentioned above, the symmetry matching principle is the first one, which determines whether atomic orbitals can be combined into molecular orbitals.The principle of energy approximation and the principle of maximum overlap of orbitals determine the efficiency of molecular orbital combination on the premise of meeting the symmetry matching principle.
Orbital energy
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It depends on the type of atomic orbit and the overlap between atomic orbits. For example, σ g1s and σ u1s are much lower than σ g2s, because the energy of atomic orbit 1s is much lower than 2s.Similarly, because of divisionhydrogen atomIn addition, the energy of 2s is significantly lower than that of 2p, so the energy of σ g2s is lower than that of σ g2p.In addition, as long as the nuclear spacing is not very small, the overlap between two 2s orbitals or two 2pz orbitals is much larger than the overlap between two 2py or 2px orbitals, so the energy difference between the bonding and anti bonding π orbitals is greater than the correspondingσ orbitThe difference is small.According to this discussion, the molecular orbital order listed in Table 2 can be expected to be: σ g1s<σ u1s<;σg2s<；σu2s<；σg2p<；πg2p<；πu2p<；σu2p ⑼
The above equation is the roughest orbit approximation, and a better approximation contains moreAtomic orbitalThese atomic orbitals meet the three conditions for effective bonding.For example, instead of pure 2s and 2pzLCAOThe σ type molecular orbital formed shall be:
⑽
c1、c2、c3、c4 The four σ orbits determined are closer to reality than the original σ g2s, σ u2s, σ g2p and σ u2p, where σ g2s and σ u2s will decrease, and σ g2p and σ u2p will increase.In addition, when the nuclear spacing becomes smaller, π u2p will decrease, leading to the possible reversal of the order of σ g2p and π u2p in Eq. 9: π u2p<;σg2p ⑾
N2 molecule belongs to this type.
With the energy level order of EqPauli principleTo predict the ground state of homonuclear diatomic molecules (Table 3).
The symbols ∑, Π,... in the table have the same meaning as those of σ, π,..., and have the meaning of angular momentum along the nuclear spacing direction, marking the state of intact moleculesOrbit determination；The+and - signs in the upper right corner indicate that the mirror surface that bisects two nuclei is symmetric or antisymmetric.
Polyatomic moleculeAbove the molecular orbital concept and method based on the approximate solution of the single electron wave equation,
Table 3
It can be naturally extended to complex polyatomic molecules.For diatomic molecules, there are angular momentum quantum numbers along the nuclear spacing directionm=0, ± 1,..., etc. to characterize the orbit or state;But for polyatomic molecules, we can't find a simple and typical molecule like H Son, so we can't solve it accurately, which causes trouble for the discussion of the problem.But since we found outQuantum numberThe symmetry of the molecular orbital represented is essentially derived from the symmetry of the molecule itself, so the symmetry analysis（group theory）It will give important information about the electronic state of any molecule without knowing the specific function of molecular orbital.The progress in this field is enormous, such as the application of group theory in chemistry,Energy level correlation diagram、Conservation principle of molecular orbital symmetryAnd so on.In addition, it is established in theElectronic energy levelTheoretical calculation methods based on orbital approximation have been developed, such as free electronsMolecular orbital method、Hull molecular orbital methodandExtended huckel molecular orbital methodEtc.
As mentioned earlier, the molecular orbital and energy level are the eigensolutions of the single electron wave equation, that is, they meet:Hψi=εψi ⑿
WhereHIt's a single electronHamiltonian operatorThe potential energy describes the average action of an electron on the fixed molecular skeleton and other electrons.Therefore,HIt is related to the motion state of other electrons, that is, the orbit.The previous discussion did not touch at allHThe specific form of the molecular orbital has not been strictly defined, and the conclusions obtained are qualitatively applicable.In order to adapt to the quantitative development of theory, the famous Hartley Fokker equation has been derived (seeSelf consistent field molecular orbital method）, for the closed shell electronic systemHIn the form of Fokker operator:
⒀
WherehIs the Hamiltonian operator of a single electron in a pure nuclear field, 2Jj-Kj=JiAnd 2Jj(j≠i）Represents the average electrostatic potential of the remaining electrons,Kj(j≠i）It is called exchange potential energy, and it comes from the correlation between electrons with the same spin caused by Pauli exclusion principle.JJ andKThe expressions of j are obviously related to molecular orbitals.
Using LCAO method, the molecular orbital ψ k is expressed as atomic orbital according to Formula ⑶φl(l=1,2，…，n）Linear combination of:
⒁
Substitute into formula 12, multiply the left and right ends byφ奰 union integral, solve the eigenvalue of the secular equationESelf consistent calculation of k and eigenvectors.Although Hartley Fokker equation carefully considered the repulsion between electrons, the average potential field model still left some inherent "correlation effects" out of consideration, so the theoretical calculation results still did not reach the accuracy of quantitative agreement with the experimental values.The way to improve is to consider configuration interaction, and many configuration interaction molecular orbitals have emergedAb initio calculationProgram for quantum chemistry research.
Application and achievements
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Many physical chemists have improved the molecular orbital theory and developed many methods to explain the structure and properties of substances by applying the molecular orbital theory.
HF method
The Hartree Fock SCF method is an ab initio calculation method, and the ab initio calculation method is simply to use a "correct" Hamiltonian operator, with the exception of the most basic constants, and no reference to any experimental dataSchrodinger equationBased on the non relativistic approximation, the method of solving Schrodinger equation and calculating molecular orbital based on the Born Oppenheimer maximum speed and single electron approximation.
Semi empirical calculation method
The semi empirical method assumes an approximateHamiltonian operator, and use various experimental data, such asionization energy, transition energy of electronic spectrumBond energyAnd other data, further simplifying the difficulty of integration. The Huckel molecular orbital theory (HMO) is a typical example of this method.
Other methods
There are also quantum chemistry composite methods, quantum Monte Carlo (QM), configuration interaction (CI), multi configuration self consistent field (MCSCF)Multibody perturbation theory, coupled clustering (CC) and other methods, which are developed based on molecular orbital theory.[2]
