11-2 The Molecular Hamiltonian
Section 11-3 The Form of the Wavefunction
11-2 The Molecular Hamiltonian In practice, one usually does not use the complete hamiltonian for an isolated molecular system. The complete hamiltonian includes nuclear and electronic kinetic energy operators, electrostatic interactions between all charged particles, and interactions between all magnetic moments due to spin and orbital motions of nuclei and electrons. Also an accounting for the fact that a moving particle experiences a change in mass due to relativistic effects is included in the complete hamiltonian. The resulting hamiltonian is much too complicated to work with. Usually, relativistic mass effects are ignored, the Born–Oppenheimer approximation is made (to remove nuclear kinetic energy operators), and all magnetic interactions are ignored (except in special cases where we are interested in spin coupling). The resulting hamiltonian for the electronic energy is, in atomic units,
n
N
n
n−1
n
ˆ
H = − 1 ∇2 − Zµ/rµi +
1/rij
(11-2)
2
i
i=1
µ=1 i=1
i=1 j =i+1 where i and j are indices for the n electrons and µ is an index for the N nuclei. The nuclear repulsion energy Vnn is
N −1
N
Vnn = ZµZν/rµν
(11-3) µ=1 ν=µ+1
In choosing this hamiltonian, we are in effect electing to seek an energy of an idealized nonexistent system—a nonrelativistic system with clamped nuclei and no magnetic moments. If we wish to make a very accurate comparison of our computed results with experimentally measured energies, it is necessary to modify either the experimental or the theoretical numbers to compensate for the omissions in ˆ H .
11-3 The Form of the Wavefunction The wavefunction for an SCF calculation is one or more antisymmetrized products of one-electron spin-orbitals. We have already seen (Chapter 5) that a convenient way to produce an antisymmetrized product is to use a Slater determinant. Therefore, we take the trial function ψ to be made up of Slater determinants containing spin-orbitals φ.
If we are dealing with an atom, then the φ’s are atomic spin-orbitals. For a molecule, they are molecular spin-orbitals.
In our discussion of many-electron atoms (Chapter 5), we noted that certain atoms in their ground states are fairly well described by assigning two electrons, one of each spin, to each AO, starting with the lowest-energy AO and working up until all the electrons are assigned. If the last electron completes the filling of all the AOs having a given principal quantum number, n, we have a closed shell atomic system. Examples are He(1s2) and Ne(1s22s22p6). Atoms wherein the last electron completes the filling of all AOs having a given l quantum number are said to have a closed subshell. An example is Be(1s22s2). Both types of system tend to be well approximated by a single determinantal wavefunction if the highest filled level is not too close in energy to the
11-2 The Molecular Hamiltonian In practice, one usually does not use the complete hamiltonian for an isolated molecular system. The complete hamiltonian includes nuclear and electronic kinetic energy operators, electrostatic interactions between all charged particles, and interactions between all magnetic moments due to spin and orbital motions of nuclei and electrons. Also an accounting for the fact that a moving particle experiences a change in mass due to relativistic effects is included in the complete hamiltonian. The resulting hamiltonian is much too complicated to work with. Usually, relativistic mass effects are ignored, the Born–Oppenheimer approximation is made (to remove nuclear kinetic energy operators), and all magnetic interactions are ignored (except in special cases where we are interested in spin coupling). The resulting hamiltonian for the electronic energy is, in atomic units,
n
N
n
n−1
n
ˆ
H = − 1 ∇2 − Zµ/rµi +
1/rij
(11-2)
2
i
i=1
µ=1 i=1
i=1 j =i+1 where i and j are indices for the n electrons and µ is an index for the N nuclei. The nuclear repulsion energy Vnn is
N −1
N
Vnn = ZµZν/rµν
(11-3) µ=1 ν=µ+1
In choosing this hamiltonian, we are in effect electing to seek an energy of an idealized nonexistent system—a nonrelativistic system with clamped nuclei and no magnetic moments. If we wish to make a very accurate comparison of our computed results with experimentally measured energies, it is necessary to modify either the experimental or the theoretical numbers to compensate for the omissions in ˆ H .
11-3 The Form of the Wavefunction The wavefunction for an SCF calculation is one or more antisymmetrized products of one-electron spin-orbitals. We have already seen (Chapter 5) that a convenient way to produce an antisymmetrized product is to use a Slater determinant. Therefore, we take the trial function ψ to be made up of Slater determinants containing spin-orbitals φ.
If we are dealing with an atom, then the φ’s are atomic spin-orbitals. For a molecule, they are molecular spin-orbitals.
In our discussion of many-electron atoms (Chapter 5), we noted that certain atoms in their ground states are fairly well described by assigning two electrons, one of each spin, to each AO, starting with the lowest-energy AO and working up until all the electrons are assigned. If the last electron completes the filling of all the AOs having a given principal quantum number, n, we have a closed shell atomic system. Examples are He(1s2) and Ne(1s22s22p6). Atoms wherein the last electron completes the filling of all AOs having a given l quantum number are said to have a closed subshell. An example is Be(1s22s2). Both types of system tend to be well approximated by a single determinantal wavefunction if the highest filled level is not too close in energy to the