XIII. The Hydrogen molecule We are now in a position to discuss the electronic structure of the simplest molecule: H2. For the low-lying electronic states of H2, the BO approximation is completely satisfactory, and so we will be interested in the electronic Hamiltonian 12221122212121111111ˆrRHBABAABel+−−−−−−−−+∇−∇−=RrRrRrRr where “1” and “2” label the two electrons and “A” and “B” label the two nuclei. a. Minimal Atomic Orbital Basis It is not possible to solve this problem analytically, and so we want to follow our standard prescription for solving this problem: we define a basis set and then crank through the linear algebra to solve the problem in that basis. Ideally, we would like a very compact basis that does not depend on the configuration of the molecule; that is, we want basis functions that do not depend on the distance between the two nuclei, ABR . This will simplify the work of doing calculations for different bond lengths. The most natural basis functions are the atomic orbitals of the individual Hydrogen atoms. If the bond length is very large, the system will approach the limit of two non-interacting Hydrogen atoms, in which case the electronic wavefunction can be well approximated by a product of an orbital on atom “A” and an orbital on atom “B” and these orbitals will be exactly the atomic orbitals (AOs) of the two atoms. Hence, the smallest basis that will give us a realistic picture of the ground state of this molecule must contain two functions: As1 and Bs1 . These two orbitals make up the minimal AO basis for H2. For finite bond lengths, it is advisable to allow the AOs to polarize and deform in response to the presence of the other electron (and the other nucleus). However, the functions we are denoting “As1 ” and“Bs1 ” need not exactly be the Hydrogenic eigenfunctions; they should look similar to the 1s orbitals, but any atom-centered functions would serve the same purpose. Since the actual form of the orbitals will vary, in what follows, we will give all the expressions in abstract matrix form, leaving the messy integration to be done once the form of the orbitals is specified. b. Molecular Orbital Picture We are now in a position to discuss the basic principles of the molecular orbital (MO) method, which is the foundation of the electronic structure theory of real molecules. The first step in any MO approach requires one to define an effective one electron Hamiltonian, effhˆ. To this end, it is useful to split the Hamiltonian into pieces for electrons “1” and “2” separately and the interaction: ( ) ( )BABAhhRrRrRrRr −−−−∇−≡−−−−∇−≡222221112121112ˆ111ˆ 12121ˆrV ≡ The full Hamiltonian is then ( ) ( )ABelRVhhH1ˆ2ˆ1ˆˆ12+++= where it should be remembered that within the BO approximation, ABR is just a number. For H2 in a minimal basis, the simplest choice for effhˆ suffices: we will choose our one electron Hamiltonian to just be the one electron part of the fullHˆ (hˆ). The matrix representation of hˆ in the minimal basis is: ≡εεABABBBABBAAAhhshsshsshsshs1ˆ11ˆ11ˆ11ˆ1. where we made use of the average one electron energy: BBAAshsshs 1ˆ11ˆ1 =≡ε and the off-diagonal coupling (often called a “resonance” integral): ABBAABshsshsh 1ˆ11ˆ1 =≡ . We can immediately diagonalize this matrix; the eigenvalues are ABh±=±εε and the eigenstates are:()()BABAssss 11112121−∝+∝−+φφ The eigenstates of the effective one-electron Hamiltonian are called molecular orbitals (just as the basis functions are called atomic orbitals). They are one-electron functions that are typically delocalized over part (or all) of the molecule. As a first step, we need to normalize these MOs. This is more complicated than it might at first appear, because the AOs are not orthogonal. For example, as the atoms approach each other, the two AOs might look like the picture at right. However, if we define the overlap integral by BAssS 11≡, we can normalize the MOs: ()( )SssssssssSssssssssBBBAABAABBBAABAA−=+−−=+=+++=−−++1111111111111111112121φφφφ which implies that the normalized wavefunctions are: ( )()( )()BASBASssss 1111121121−=+=−−++φφ. These eigenfunctions merely reflect the symmetry of the molecule; the two hydrogen atoms are equivalent and so the eigenorbitals must give equal weight to each 1s orbital. So our “choice” of the one electron Hamiltonian actually does not matter much in this case; any one-electron Hamiltonian that reflects the symmetry of the molecule will give the same molecular orbitals. For historical reasons, +φ is usually denoted σ while −φ is denoted *σ. The second step in MO theory is to construct a single determinant out of the MOs that corresponds to the state we are interested in. For the purposes of illustration, let us look at the lowest singlet state built out of the molecular orbitals. First, note that 0<ABh , so σ is lower in energy than *σ. Neglecting the interaction, then, the lowest singlet state is: σσ=ΦMO HA HBand this is the MO ground state for H2. How good an approximation is it? Well, we can compute the expectation value of the energy, σσσσelHˆ as follows. First, we decompose the wavefunction into spatial and spin parts and note that the spin part is normalized: ()()()()( ) ( ) ( ) ( )21ˆ2121ˆ21ˆσσσσσσσσσσσσelspinspinelelHHH=ΦΦ= Then, we note that ()()ABelRVhhH /1ˆ2ˆ1ˆˆ12+++= and ()()()()()()()()()()( ) ( ) ( )σεσσσσσσσσσσ≡==11ˆ12211ˆ1211ˆ21hhh ()()()()()()()()()()( ) ( ) ( )σεσσσσσσσσσσ≡==22ˆ21122ˆ2212ˆ21hhh ()()()()σσσσσσJV ≡21ˆ2112 Taken together, these facts allow us to write: ABABMOMOMOMOMOMOMOMORJRVhhh121ˆˆˆˆ12211++=+ΨΨ+ΨΨ+ΨΨ=ΨΨσσσε Each of the first two terms is energy of a single electron (either 1 or 2) in the field produced by the nuclei (hˆ) while the third is the average repulsion of the two electrons. Note that the second and third terms are both positive, so binding has to arise from the one-electron piece. This is the MO energy for the ground state of H2. For a reasonable choice of the 1s-like basis functions – it turns out to be more convenient to fit the exponential decay of the hydrogenic orbitals to a sum of Gaussians- we can use a computer to compute the unknowns above (σε and σσJ ) and plot