Section 11-6
Section 11-6 Interpretation of the LCAO-MO-SCF Eigenvalues
ˆ
Kj leads ultimately to the production of exchange integrals, and so it is called an exchange operator. It is written explicitly in conjunction with a function on which it is operating, viz.
ˆ Kj φi(1) = φ∗j (2) (1/r12)φi(2)dτ (2) φj (1)
(11-9)
Notice that an index exchange has been performed. It is not difficult to see that the expression (see Appendix 9 for bra-ket notation) φi| ˆF|φi = i
(11-10)
will lead to integrals such as φi| ˆJj |φi = φi (1) φj (2) |1/r12|φi (1) φj (2) = Jij
(11-11)
φi| ˆ Kj |φi = φi (1) φj (2) |1/r12|φi (2) φj (1) = Kij
(11-12)
which are formally the same as the coulomb and exchange terms encountered in Chapter 5 in connection with the helium atom. Notice that, if the spins associated with spin-orbitals φi and φj differ, Kij must vanish. This arises because integrations over space and spin coordinates of electron 1 (or 2) in Eq. (11-12) lead to integration over two different (and orthogonal) spin functions. On the other hand, Jij is not affected by such spin agreement or disagreement.
It would appear from Eq. (11-6) that the MOs φ are eigenfunctions of the Fock operator and that the Fock operator is, in effect, the hamiltonian operator. There is an important qualitative difference between ˆ F and ˆ H , however. The Fock operator is itselfa function of the MOs φ. Since the summation index j in Eq. (11-7) includes i, the operators ˆ Ji and ˆ Ki must be known in order to write down ˆ F , but ˆ Ji and ˆ Ki involve φi, and φi is an eigenfunction of ˆ F . Hence, we need ˆ F to find φi, and we need φi to know ˆF. To circumvent this problem, an iterative approach is used. One makes an initial guess at the MOs φ. (One could use a semiempirical method to produce this starting set.) Then these MOs are used to construct an operator ˆ F , which is used to solve for the new MOs φ. These are then used to construct a new Fock operator, which is in turn used to find new MOs, which are used for a new ˆ F , etc., until at last no significant change is detected in two successive steps of this procedure. At this point, the φ’s produced by ˆ F are the same as the φ’s that produce the coulomb-and-exchange fields in ˆ F . The solutions are said to be self-consistent, and the method is referred to as the self-consistent-field (SCF) method.
11-6 Interpretation of the LCAO-MO-SCF Eigenvalues
The physical meaning of an eigenvalue i is best understood by expanding the integral i = φi| ˆ
F |φi
(11-13)
with ˆ F given by Eq. (11-7). We obtain
n
i = φi − 1 ∇2 φi Zu/rµ1 φi + 2Jij − Kij
(11-14)
2 1 φi − µ j =1
Chapter 11 The SCF-LCAO-MO Method and Extensions
It is common practice to combine the first two terms of Eq. (11-14), which depend only on the nature of φi, into a single expectation value of the one-electron part of the hamiltonian, symbolized Hii. Thus,
n
i = Hii + (2Jij − Kij )
(11-15)
j =1 The quantity Hii is the average kinetic plus nuclear-electronic attraction energy for the electron in φi.
The sum of coulomb and exchange integrals in Eq. (11-15) contains all the electronic interaction energy. Observe that the index j runs over all the occupied MOs. For a particular value of j , say j = k = i, this gives 2Jij − Kij as an interaction energy. This means that an electron in φi, experiences an interaction energy with the two electrons in φk of 2 φi(1)φk(2)|1/r12|φi(1)φk(2) − φi(1)φk(2) |1/r12| φk(1)φi(2)
(11-16)
The first part is the classical repulsion between the electron having an orbital charge cloud given by |φi|2 and the two electrons having charge cloud |φk|2. The second part is the exchange term which, as we saw in Chapter 5, arises from the antisymmetric nature of the wavefunction. It enters (11-16) only once because the electron in φi, agrees in spin with only one of the two electrons in φk, [Equation (11-15) applies because we have restricted our discussion to closed-shell systems.]
The summation over 2Jij − Kij includes the case j = i. Here we get 2Jii − Kii.
However, examination of Eqs. (11-11) and (11-12) shows that Jii = Kii, and so we are left with Jii. This corresponds to the repulsion between the electron in φi (the energy of which we are calculating) and the other electron in φi. Because these electrons must occur with opposite spin, there is no exchange energy for this interaction.
In brief, then, the quantity i, often referred to as an orbital energy or a one-electronenergy, is to be interpreted as the energy of an electron in φi, resulting from its kinetic energy, its energy of attraction for the nuclei, and its repulsion and exchange energies due to all the other electrons in their charge clouds |φj |2.
11-7 The SCF Total Electronic Energy
It is natural to suppose that the total electronic energy is merely the sum of the oneelectron energies, but this is not the case in SCF theory. Consider a two-electron system.
The energy of electron 1 includes its kinetic and nuclear attraction energies and its repulsion and exchange energies for electron 2. The energy of electron 2 includes its kinetic and nuclear attraction energies and its repulsion and exchange energies for electron 1. If we sum these, we have accounted properly for kinetic and nuclear attraction energies, but we have included the interelectronic interactions twice as much as they actually occur. (The energy of repulsion, say, between two charged particles, 1 and 2, is given by the repulsion of 1 for 2 or of 2 for 1, but not by the sum of these.) Therefore, if we sum one-electron energies, we get the total electronic energy plus an extra measure
ˆ
Kj leads ultimately to the production of exchange integrals, and so it is called an exchange operator. It is written explicitly in conjunction with a function on which it is operating, viz.
ˆ Kj φi(1) = φ∗j (2) (1/r12)φi(2)dτ (2) φj (1)
(11-9)
Notice that an index exchange has been performed. It is not difficult to see that the expression (see Appendix 9 for bra-ket notation) φi| ˆF|φi = i
(11-10)
will lead to integrals such as φi| ˆJj |φi = φi (1) φj (2) |1/r12|φi (1) φj (2) = Jij
(11-11)
φi| ˆ Kj |φi = φi (1) φj (2) |1/r12|φi (2) φj (1) = Kij
(11-12)
which are formally the same as the coulomb and exchange terms encountered in Chapter 5 in connection with the helium atom. Notice that, if the spins associated with spin-orbitals φi and φj differ, Kij must vanish. This arises because integrations over space and spin coordinates of electron 1 (or 2) in Eq. (11-12) lead to integration over two different (and orthogonal) spin functions. On the other hand, Jij is not affected by such spin agreement or disagreement.
It would appear from Eq. (11-6) that the MOs φ are eigenfunctions of the Fock operator and that the Fock operator is, in effect, the hamiltonian operator. There is an important qualitative difference between ˆ F and ˆ H , however. The Fock operator is itselfa function of the MOs φ. Since the summation index j in Eq. (11-7) includes i, the operators ˆ Ji and ˆ Ki must be known in order to write down ˆ F , but ˆ Ji and ˆ Ki involve φi, and φi is an eigenfunction of ˆ F . Hence, we need ˆ F to find φi, and we need φi to know ˆF. To circumvent this problem, an iterative approach is used. One makes an initial guess at the MOs φ. (One could use a semiempirical method to produce this starting set.) Then these MOs are used to construct an operator ˆ F , which is used to solve for the new MOs φ. These are then used to construct a new Fock operator, which is in turn used to find new MOs, which are used for a new ˆ F , etc., until at last no significant change is detected in two successive steps of this procedure. At this point, the φ’s produced by ˆ F are the same as the φ’s that produce the coulomb-and-exchange fields in ˆ F . The solutions are said to be self-consistent, and the method is referred to as the self-consistent-field (SCF) method.
11-6 Interpretation of the LCAO-MO-SCF Eigenvalues
The physical meaning of an eigenvalue i is best understood by expanding the integral i = φi| ˆ
F |φi
(11-13)
with ˆ F given by Eq. (11-7). We obtain
n
i = φi − 1 ∇2 φi Zu/rµ1 φi + 2Jij − Kij
(11-14)
2 1 φi − µ j =1
Chapter 11 The SCF-LCAO-MO Method and Extensions
It is common practice to combine the first two terms of Eq. (11-14), which depend only on the nature of φi, into a single expectation value of the one-electron part of the hamiltonian, symbolized Hii. Thus,
n
i = Hii + (2Jij − Kij )
(11-15)
j =1 The quantity Hii is the average kinetic plus nuclear-electronic attraction energy for the electron in φi.
The sum of coulomb and exchange integrals in Eq. (11-15) contains all the electronic interaction energy. Observe that the index j runs over all the occupied MOs. For a particular value of j , say j = k = i, this gives 2Jij − Kij as an interaction energy. This means that an electron in φi, experiences an interaction energy with the two electrons in φk of 2 φi(1)φk(2)|1/r12|φi(1)φk(2) − φi(1)φk(2) |1/r12| φk(1)φi(2)
(11-16)
The first part is the classical repulsion between the electron having an orbital charge cloud given by |φi|2 and the two electrons having charge cloud |φk|2. The second part is the exchange term which, as we saw in Chapter 5, arises from the antisymmetric nature of the wavefunction. It enters (11-16) only once because the electron in φi, agrees in spin with only one of the two electrons in φk, [Equation (11-15) applies because we have restricted our discussion to closed-shell systems.]
The summation over 2Jij − Kij includes the case j = i. Here we get 2Jii − Kii.
However, examination of Eqs. (11-11) and (11-12) shows that Jii = Kii, and so we are left with Jii. This corresponds to the repulsion between the electron in φi (the energy of which we are calculating) and the other electron in φi. Because these electrons must occur with opposite spin, there is no exchange energy for this interaction.
In brief, then, the quantity i, often referred to as an orbital energy or a one-electronenergy, is to be interpreted as the energy of an electron in φi, resulting from its kinetic energy, its energy of attraction for the nuclei, and its repulsion and exchange energies due to all the other electrons in their charge clouds |φj |2.
11-7 The SCF Total Electronic Energy
It is natural to suppose that the total electronic energy is merely the sum of the oneelectron energies, but this is not the case in SCF theory. Consider a two-electron system.
The energy of electron 1 includes its kinetic and nuclear attraction energies and its repulsion and exchange energies for electron 2. The energy of electron 2 includes its kinetic and nuclear attraction energies and its repulsion and exchange energies for electron 1. If we sum these, we have accounted properly for kinetic and nuclear attraction energies, but we have included the interelectronic interactions twice as much as they actually occur. (The energy of repulsion, say, between two charged particles, 1 and 2, is given by the repulsion of 1 for 2 or of 2 for 1, but not by the sum of these.) Therefore, if we sum one-electron energies, we get the total electronic energy plus an extra measure