Calculation of energy of molecular orbital from wave function
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The Hydrogen Molecule-Ion
Molecular orbital theory is a conceptual extension of the orbital model, which was so successfully applied to atomic structure. As was once playfully remarked, "a molecule is nothing more than an atom with more nuclei." This may be overly simplistic, but we do attempt, as far as possible, to exploit analogies with atomic structure. Our understanding of atomic orbitals began with the exact solutions of a prototype problem – the hydrogen atom. We will begin our study of homonuclear diatomic molecules beginning with another exactly solvable prototype, the hydrogen molecule-ion H+2H2+ . This species actually has a transient existence in electrical discharges through hydrogen gas and has been detected by mass spectrometry. It also has been detected in outer space. The Schrödinger equation for H +22+ can be solved exactly within the Born-Oppenheimer approximation. For fixed internuclear distance R, this reduces to a problem of one electron in the field of two protons, designated A and B. We can write
{−12∇21rA−1rB+1R}ψ(r)=Eψ(r)(1)
(1){−12∇21rA−1rB+1R}ψ(r)=Eψ(r)
where rA and rB are the distances from the electron to protons A and B, respectively. This equation was solved by Burrau (1927), after separating the variables in prolate spheroidal coordinates.
Molecular Orbitals Involving Only ns Atomic Orbitals
We begin our discussion of molecular orbitals with the simplest molecule, H2, formed from two isolated hydrogen atoms, each with a 1s11s1 electron configuration. As discussed before, electrons can behave like waves. In the molecular orbital approach, the overlapping atomic orbitals are described by mathematical equations called wave functions. The 1s atomic orbitals on the two hydrogen atoms interact to form two new molecular orbitals, one produced by taking the sum of the two H 1s wave functions, and the other produced by taking their difference:
MO(1)=MO(1)=AO(atomA)AO(atomA)+−AO(atomB)AO(atomB)(5.5.1)
(5.5.1)MO(1)=AO(atomA)+AO(atomB)MO(1)=AO(atomA)−AO(atomB)
The molecular orbitals created from the sum and the difference of two wavefunctions (atomic orbitals) from Equation 5.5.15.5.1 are called Linear Combinations of Atomic Orbitals (LCAOs). A molecule must have as many molecular orbitals as there are atomic orbitals.
Adding two atomic orbitals corresponds to constructive interference between two waves, thus reinforcing their intensity; the internuclear electron probability density is increased. The molecular orbital corresponding to the sum of the two H 1s orbitals is called a σ1s combination (pronounced “sigma one ess”) (part (a) and part (b) in Figure 5.5.15.5.1 ). In a sigma (σ) orbital, A bonding molecular orbital in which the electron density along the internuclear axis and between the nuclei has cylindrical symmetry, the electron density along the internuclear axis and between the nuclei has cylindrical symmetry; that is, all cross-sections perpendicular to the internuclear axis are circles. The subscript 1s denotes the atomic orbitals from which the molecular orbital was derived: The ≈ sign is used rather than an = sign because we are ignoring certain constants that are not important to our argument.
Figure 5.5.15.5.1 : Molecular Orbitals for the H2 Molecule. (a) This diagram shows the formation of a bonding σ1s molecular orbital for H2 as the sum of the wave functions (Ψ) of two H 1s atomic orbitals. (b) This plot of the square of the wave function (Ψ2) for the bonding σ1s molecular orbital illustrates the increased electron probability density between the two hydrogen nuclei. (Recall that the probability density is proportional to the square of the wave function.) (c) This diagram shows the formation of an antibonding σ∗1sσ1s∗ molecular orbital for H2 as the difference of the wave functions (Ψ) of two H 1s atomic orbitals. (d) This plot of the square of the wave function (Ψ2) for the σ∗1sσ1s∗ antibonding molecular orbital illustrates the node corresponding to zero electron probability density between the two hydrogen nuclei.
σ1s≈1s(A)+1s(B)(5.5.2)
(5.5.2)σ1s≈1s(A)+1s(B)
Conversely, subtracting one atomic orbital from another corresponds to destructive interference between two waves, which reduces their intensity and causes a decrease in the internuclear electron probability density (part (c) and part (d) in Figure 5.5.15.5.1 ). The resulting pattern contains a node where the electron density is zero. The molecular orbital corresponding to the difference is called σ∗1sσ1s∗ (“sigma one ess star”). In a sigma star (σ*) orbital An antibonding molecular orbital in which there is a region of zero electron probability (a nodal plane) perpendicular to the internuclear axis., there is a region of zero electron probability, a nodal plane, perpendicular to the internuclear axis:
σ⋆1s≈1s(A)−1s(B)(5.5.3)
(5.5.3)σ1s⋆≈1s(A)−1s(B)
A molecule must have as many molecular orbitals as there are atomic orbitals.