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Many-electron wave functions and transi-tion moments
The goal of quantum chemistry is to find eigenfunctions to the electronic Hamiltonian and use them to calculate molecular properties. This requires a formalism for the electronic wave functions , which may represent hundreds of electrons in large molecules. Such a formalism is introduced in this section.
For Hamiltonians which do not contain any electron–electron interactions, the Schrödinger equation becomes separable in the electron coordinates and its solutions are antisymmetrized products of one-electron wave functions. In general these products can be written as Slater determinants, and their form will be introduced shortly. The one-electron wave functions making up the Slater determinants are eigenfunctions of the one-electron operators constituting the no-interaction-Hamiltonian. The total state is then deter-mined by the one-particle states occupied by the electrons. The one-particle states are termed orbitals.
Electrons have spin and two electrons of opposite spin can share orbitals with identical spatial extents. Thus there is a distinction made between spatial orbitals, dependent only on the three spatial coordinates r and written (r), and spin orbitals, which depend on r and one symbolic spin coordinate !. The spin orbitals are written (x), where x = (r; !). All operators which do not act on spin have spatial orbitals (r) as their eigenfunctions. In those cases, spin orbitals may be constructed from spatial orbitals via multipli-cation by a spinor (!): (x) = (r) (!). The spatial orbitals (r) can generally be ordered by increasing energy and indexed i. Similar orderings are used to index the spin orbitals i, so that the N lowest indices 0 : : : N correspond to the N spin orbitals with lowest energy. If the Hamiltonian lacks spin-dependent operators, the ordering of spin orbitals will necessar-ily be degenerate. It is then common to employ some systematic indexing scheme, like making all spin orbitals with odd index be spin up and all with even index be spin down.
When electron-electron interactions are included in the Hamiltonian, the Hamiltonian is formally no longer separable, and Slater determinants can-not be its eigenfunctions. However, it is useful to keep the picture of the electronic wave function being made up of individual spin orbitals as a still valid description. The reason is that given a set of spin orbitals which completely spans the one-electron wave function space, the set of Slater deter-minants which can be constructed from those spin orbitals will be a complete set in the N-electron wave function space, and the exact eigenfunctions of the Hamiltonian will then be expandable in the Slater determinants. Addi-tionally, the Hartree–Fock-method, which will be introduced later, provides a way to find the Slater determinant most closely approximating the exact wave function, and if the exact ground state wave function is expanded in the basis of Slater determinants constructed from spin orbitals obtained via the Hartree–Fock method, the ground state Hartree–Fock Slater determinant will have a dominating coefficient, meaning its separable description of the exact wave function is accurate to a good degree.
Hartree–Fock method
As mentioned, since the electronic Hamiltonian He contains electron–electron interaction terms, its eigenfunctions are not of Slater determinant form. How-ever, an interesting question is which Slater determinant most closely approx-imates the ground state of the electronic Hamiltonian. Finding that Slater determinant is done by finding a set of spin orbitals from which it can be built. From the same spin orbitals excited determinants can also be built, and they can then be used to expand the ground state in a best-approximating basis.
The best-approximating Slater determinant can be found via the variational principle, which yields the eigenvalue equation « h(x1) + 2 N dx2 b (x2)r121(1 P12) b(x2)# a(x1) 1 Z a Xf(x1) a(x1) = « a a(x1) (2.12).
where the expression in brackets is defined as the Fock operator f(x). This equation is called the Hartree–Fock equation and the spin orbitals a satis-fying it are called the Hartree–Fock spin orbitals. Some important details of the derivation of the Hartree–Fock equation are given in Appendix A. The full derivation is given by Szabo and Ostlund.11 The eigenvalue « a is the orbital energy of a. Finding the eigenfunctions to the Fock operator then yields the spin orbitals from which the best-approximating Slater de-terminants may be constructed. The Slater determinant containing the N Hartree–Fock spin orbitals with the lowest energies is called the Hartree-Fock ground state determinant.
Time-dependent perturbation theory
Electronic structure theory provides approximate solutions to the time-independent Schrödinger equation, but the interaction of molecules with light is time-dependent, and is described by the time-dependent Schrödinger equationH @t :^^ = i~ @ (3.1).
where H is now time-dependent. To find time-dependent properties of molecules, approximate methods to solve (3.1) are required. If the time-dependent in-teraction is small, it may be treated as a perturbation. The details follow. ^^ ^ ^ ^ is a time-independent Hamiltonian, like.
Let H be H = H0 + V , where H0 the molecular Hamiltonian or the Hartree–Fock Hamiltonian. Then there is ^ fj ig a complete set of eigenfunctions to H0, n , which in the case of molecules are found by the Hartree–Fock method or DFT (as explained in Chapter 2) and can be taken to be sums of Slater determinants. Then the eigenfunctions are solutions to (3.1) when combined with an exponential factor: jni e iEnt=~.
Proofs of one-electron operator expressions
We are interested in calculating the observable A (which, to exemplify, could be the electric or magnetic dipole moment) for Slater determinants j i = ^ N i jk By definition, A is represented by the operator A = Pi=1 a^(xi).
Insertionj ofi.A^ into the expectation value h jA^j i gives N N a^(i) j i = h j a^(i) j i h jAj i = h j i=1 =1 ^ X Xi (A.1).
Since each wave function k appears as a function of the coordinates xi in the determinant j i, all the terms in the sum (A.1) are in fact identical. The actual index used for the coordinate of the operator a^ is also unimportant since it is integrated over. Thus the sum is equal to N h j a^ j i. To go further we must consider the form of the Slater determinants j i. To be general, we will allow the determinants to differ in wave function composition. Using the ground state j 0i = j 1 2 N i as starting point, note that a general form to write an N N-determinant is 1 N! j 0i = pN! =1 Xi ( 1)pi Pif 1 2 N g (A.2).
Table of contents :
1 Introduction
2 Electronic Structure Theory
2.1 Molecular Hamiltonians
2.2 Many-electron wave functions and transition moments
2.3 Hartree–Fock method
2.4 Density functional theory
3 Time-Dependent Molecular Properties
3.1 Time-dependent perturbation theory
3.2 Circular dichroism
4 Investigation
4.1 Computational details
4.2 Results and Discussion
4.3 Conclusion
References
A Some extra details of electronic structure theory
A.1 Proofs of one-electron operator expressions
A.2 Considerations when deriving the Hartree–Fock equation . . .
