Due to the double slit experiment, we are already familiar with one fundamental principle of quantum mechanics, i.e. that the true probability amplitude for a system can be described as a linear combination of more fundamental single amplitudes. Now we try to determine molecular energy levels and wave functions. Therefore, we need to solve the Schroedinger equation Hψ_{n} = E_{n} ψ_{n}, with the molecule's Hamiltonian operator H and ψ_{n} as a representation of the molecular orbitals. For analyzing the system, we expand the eigenfunction ψ_{n} by employing well known fundamental functions φ_{k}. It is understood that these functions φ_{k} are no eigenfunctions with respect to Hψ_{n}: y = ∑_{k} c_{k}φ_{k} (to simplify, index n has been omitted here). The following explanations treat the problem in an abstract mathematical manner. Anyone who wishes getting a first impression of the principles is invited to click here.
The most conspicious and  when compared with atomic orbitals  distinctive property is that molecular orbitals are polycentric. The best way of describing an electron is to state that it moves within an orbital that is located around both (respectively all) nuclei of the molecule's atoms. Now we want to describe molecular orbitals ψ as a linear combination of atomic orbitals φ (LCAO = Linear Combination of AtomicOrbitals). The appropriate φ is soon to be presented. In a general treatment, first we expand the function ψ for any molecule using functions φ and use the common abbreviations. The equations in the left column refer to the general case where the number of functions φ remains open, whereas the equations in the right column refer to diatomic molecules for which two functions φ are employed.


ψ = ∑_{k} c_{k}φ_{k}  ψ = c_{A}φ_{A} + c_{B}φ_{B} 


Multiplication with φ_{i}* and ∫:  Multiplication with φ_{A}* and ∫: 


S_{ik} = ∫φ_{i}*φ_{k}dτ 
S_{AB} = ∫φ_{A}*φ_{B} 

c_{A}(H_{BA} − E·S_{BA}) + c_{B} (H_{BB}−E) = 0 
We recieved a linear system of equations, the socalled "secular equations". Besides the trivial case with c_{k} = 0, there is only one solution where the secular determinant disappears. For the diatomic molecule, the problem is easily solved by transforming both equations to have c_{A}/c_{B} on the left side and equate both equations of the system. Maybe it is confusing to obtain the term ∫φ_{i}*φ_{k}dτ as we have learned that the functions, i.e. the atomic orbitals, are orthogonal. This still holds true, but in contrast to AOs of single atoms, in our case, each AO has a centre of its own because there is a shift in position. In consequence, they are not necessarily orthogonal.
We already anticipate that it would be desirable to know the case where the integral ∫ φ_{A}*φ_{B}dτ disappears. Anyone who wants to consider the problem for himself in advance may combine an atomic orbital of atom A, e.g. 1s_{A} with atomic orbitals like 2p_{x}, 2p_{y} or 2p_{z} of atom B and pay attention to the case where the integral becomes zero. But be aware that H_{AA}, H_{AB}, S_{AB} etc. appear in all applications of the LCAOMOtheory and thus it is indispensible to know their physical meaning. Often, you will find another representation:
α_{A} = H_{AA} = ∫φ_{A}* Hφ_{A}dτ
α_{B} = H_{BB} = ∫φ_{B}*
Hφ_{B}dτ
β = H_{AB} = H_{BA} = ∫φ_{A}*
Hφ_{B}dτ
S_{BA} = S_{AB} = S = ∫φ_{A}*φ_{B}dτ
Using this notation, the equations for the ABsystem become:
c_{A}(α_{A}−E) + c_{B}(β−E·S) = 0
c_{A}(β−E·S) + c_{B}(α_{B}−E) = 0
First, let's focus on the meaning of involved quantities: Under the assumption of a normalized function φ_{A} the α_{A} represents roughly the energy of an electron imagined to move in orbital φ_{A} of atom A; an analogous interpretation applies to α_{B}. Energy α_{A} is below E_{A} for the wave function φ_{A} of an electron bound solely to nucleus A because in H, there is also a component that represents attraction to nucleus B. α_{A} is the expected value for the energy that is described by φ_{A}. Especially the sphere close to nucleus A contributes to α_{A} where φ_{A} is large too, and this is as well the sphere where H resembles the operator of the unbound atom. Thus α_{A} (oder α_{B}) is approximately the energy of an electron in φ_{A} (or φ_{B}) close to nucleus A (or B) which is slightly changed by the presence of the other atom. As α_{A} and α_{B} contain terms for Coulomb's forces, they are called the coulombic integral. For a certain type of orbital, the absolute value of α increases from the left to the right for the elements of one row in the periodic table of elements. Note that the same increase is found for the atom's electronegativity or "attraction" towards electrons.
The overlap S_{12} between two identical sorbitals φ_{1} and φ_{2}: (a) S_{12} ≈ 0; (b) small S_{12}; (c) S_{12} →1. 
The integrals above depend both on φ_{A} and φ_{B}. Obviously, the values for β and S are strongly dependent on possible overlap between the two atomic orbitals. S is called the overlap integral and β resonance integral . The solution of the secular equations for homonuclear molecules (consisting identical atoms A and B) with α_{A} = α_{B} differs from that for heteronuclear molecules (α_{A} ≠α_{B}). Thus, we treat the two cases independently. To comprehend the essence, we first focus on the most simple molecule  the H_{2}^{+}Ion. Maybe, the laboratory chemist is disappointed now as he never faces this ion. But even he won't be able to treat complicated molecules without an understanding for the basic ones.
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