less strongly with the size of the ao basis a. Koopmans'Theorem The HF-SCF equations he pi=Ei i imply that the orbital enegies Ei can be Ei=<h|4>=<中T+Ⅴ|>+ j(occupied)<中lJ-Kjl> ≤T+V|>+ occupied)[J-K where T+ V represents the kinetic (T)and nuclear attraction()energies, respectively Thus, Ei is the average value of the kinetic energy plus Coulombic attraction to the nuclei for an electron in i plus the sum over all of the spin-orbitals occupied inYof Coulomb minus exchange interactions If oi is an occupied spin-orbital, the j=i term[ Ji i-Ki.i] disappears in the above sum and the remaining terms in the sum represent the Coulomb minus exchange interaction of i wit all of the N-1 other occupied spin-orbitals. If oi is a virtual spin- orbital, this cancellation does not occur because the sum over j does not include j=i. So one obtains the Coulomb minus exchange interaction of i with all n of the occupied spin-orbitals in p. Hence the energies of occupied orbitals pertain to interactions appropriate to a total of n electrons, while the energies of virtual orbitals pertain to a system with N+l electrons Let us consider the following model of the detachment or attachment of an 21
21 less strongly with the size of the AO basis. a. Koopmans’ Theorem The HF-SCF equations he fi = ei fi imply that the orbital enegies ei can be written as: ei = < fi | he | fi > = < fi | T + V | fi > + Sj(occupied) < fi | Jj - Kj | fi > = < fi | T + V | fi > + Sj(occupied) [ Ji,j - Ki,j ], where T + V represents the kinetic (T) and nuclear attraction (V) energies, respectively. Thus, ei is the average value of the kinetic energy plus Coulombic attraction to the nuclei for an electron in fi plus the sum over all of the spin-orbitals occupied in Y of Coulomb minus exchange interactions. If fi is an occupied spin-orbital, the j = i term [ Ji,i - Ki,i] disappears in the above sum and the remaining terms in the sum represent the Coulomb minus exchange interaction of fi wit all of the N-1 other occupied spin-orbitals. If fi is a virtual spinorbital, this cancellation does not occur because the sum over j does not include j = i. So, one obtains the Coulomb minus exchange interaction of fi with all N of the occupied spin-orbitals in Y. Hence the energies of occupied orbitals pertain to interactions appropriate to a total of N electrons, while the energies of virtual orbitals pertain to a system with N+1 electrons. Let us consider the following model of the detachment or attachment of an
electron in an N-electron system 1. In this model, both the parent molecule and the species generated by adding or removing an electron are treated at the single-determinant level 2. The Hartree-Fock orbitals of the parent molecule are used to describe both species. It is said that such a model neglects 'orbital relaxation(i. e, the reoptimization of the spin- orbitals to allow them to become appropriate to the daughter species) Within this model, the energy difference between the daughter and the parent can be written as follows( ok represents the particular spin-orbital that is added or removed) for electron detachment and for electron attachment So. within the limitations of the hf. frozen -orbital model the ionization potentials(IPs)and electron affinities(EAs)are given as the negative of the occupied and virtual spin-orbital energies, respectively. This statement is referred to as Koopmans theorem; it is used extensively in quantum chemical calculations as a means of estimating IPs and EAs and often yields results that are qualitatively correct (i. e, +0.5 ev) b Orbital Energies and the Total energy
22 electron in an N-electron system. 1. In this model, both the parent molecule and the species generated by adding or removing an electron are treated at the single-determinant level. 2. The Hartree-Fock orbitals of the parent molecule are used to describe both species. It is said that such a model neglects 'orbital relaxation' (i.e., the reoptimization of the spinorbitals to allow them to become appropriate to the daughter species). Within this model, the energy difference between the daughter and the parent can be written as follows (fk represents the particular spin-orbital that is added or removed): for electron detachment: EN-1 - EN = - ek ; and for electron attachment: EN - EN+1 = - ek . So, within the limitations of the HF, frozen-orbital model, the ionization potentials (IPs) and electron affinities (EAs) are given as the negative of the occupied and virtual spin-orbital energies, respectively. This statement is referred to as Koopmans’ theorem; it is used extensively in quantum chemical calculations as a means of estimating IPs and EAs and often yields results that are qualitatively correct (i.e., ± 0.5 eV). b. Orbital Energies and the Total Energy
The total HF-SCF electronic energy can be written as E- Ei(occupied)pil T+ VIpi>+ 2i>j(occupied)[Jij-KijI and the sum of the orbital energies of the occupied spin-orbitals is given by 2i(occupied)Ei- >i(occupied )<pi IT+ vipi>+ 2i,j(occupied )[i, j-KijI These two expressions differ in a very important way; the sum of occupied orbital energies double counts the Coulomb minus exchange interaction energies. Thus, within the Hartree-Fock approximation, the sum of the occupied orbital energies is not equal to the total energy. This finding teaches us that we can not think of the total electronic energy of a given orbital occupation in terms of the orbital energies alone. We need to also keep track of the inter-electron Coulomb and exchange energies 5. Molecular orbitals Before moving on to discuss methods that go beyond the hf model, it is appropriate to examine some of the computational effort that goes into carrying out an SCF calculation on molecules. The primary differences that appear when molecules rather than atoms are considered are i. The electronic Hamiltonian he contains not only one nuclear-attraction Coulomb potential 2, Ze/r, but a sum of such terms, one for each nucleus in the molecule
23 The total HF-SCF electronic energy can be written as: E = Si(occupied) < fi | T + V | fi > + Si>j(occupied) [ Ji,j - Ki,j ] and the sum of the orbital energies of the occupied spin-orbitals is given by: Si(occupied) ei = Si(occupied) < fi | T + V | fi > + Si,j(occupied) [Ji,j - Ki,j ]. These two expressions differ in a very important way; the sum of occupied orbital energies double counts the Coulomb minus exchange interaction energies. Thus, within the Hartree-Fock approximation, the sum of the occupied orbital energies is not equal to the total energy. This finding teaches us that we can not think of the total electronic energy of a given orbital occupation in terms of the orbital energies alone. We need to also keep track of the inter-electron Coulomb and exchange energies. 5. Molecular Orbitals Before moving on to discuss methods that go beyond the HF model, it is appropriate to examine some of the computational effort that goes into carrying out an SCF calculation on molecules. The primary differences that appear when molecules rather than atoms are considered are i. The electronic Hamiltonian he contains not only one nuclear-attraction Coulomb potential Sj Ze2 /rj but a sum of such terms, one for each nucleus in the molecule:
2,2 Ze/r-RI, whose locations are denoted Ra ii. One has ao basis functions of the type discussed above located on each nucleus of the molecule. These functions are still denoted (r-R ) but their radial and angular dependences involve the distance and orientation of the electron relative to the particular nucleus on which the ao is located Other than these two changes, performing a SCF calculation on a molecule(or molecular ion) proceeds just as in the atomic case detailed earlier. Let us briefly review how this Iterative process occurs Once atomic basis sets have been chosen for each atom the one-and two-electron integrals appearing in the h and overlap matrices must be evaluated There are numerous highly efficient computer codes that allow such integrals to be computed for s, p, d, f, and even g, h, and i basis functions. After executing one of these 'integral packages for a basis with a total of M functions, one has available (usually on the computer's hard disk) of the order of M2/2 one-electron(<xu xv>and<xulxv>)and M4/8 two- electron(<u xo Ixylx >)integrals. When treating extremely large atomic orbital basis sets(e. g, 500 or more basis functions), modern computer programs calculate the requisite integrals but never store them on the disk. Instead, their contributions to the < xuhlxy matrix elements are accumulated 'on the fly after which the integrals are discarded a Shapes, Sizes, and Energies of Orbitals Each molecular spin-orbital(MO) that results from solving the HF SCF equations 4
24 Sa Sj Zae 2 /|rj -Ra |, whose locations are denoted Ra . ii. One has AO basis functions of the type discussed above located on each nucleus of the molecule. These functions are still denoted cm (r-Ra ), but their radial and angular dependences involve the distance and orientation of the electron relative to the particular nucleus on which the AO is located. Other than these two changes, performing a SCF calculation on a molecule (or molecular ion) proceeds just as in the atomic case detailed earlier. Let us briefly review how this iterative process occurs. Once atomic basis sets have been chosen for each atom, the one- and two-electron integrals appearing in the he and overlap matrices must be evaluated. There are numerous highly efficient computer codes that allow such integrals to be computed for s, p, d, f, and even g, h, and i basis functions. After executing one of these 'integral packages' for a basis with a total of M functions, one has available (usually on the computer's hard disk) of the order of M2/2 one-electron (< cm | he | cn > and < cm | cn >) and M4/8 twoelectron (< c m c d | c n c k >) integrals. When treating extremely large atomic orbital basis sets (e.g., 500 or more basis functions), modern computer programs calculate the requisite integrals but never store them on the disk. Instead, their contributions to the <cm |he |cn> matrix elements are accumulated 'on the fly' after which the integrals are discarded. a. Shapes, Sizes, and Energies of Orbitals Each molecular spin-orbital (MO) that results from solving the HF SCF equations
for a molecule or molecular ion consists of a sum of components involving all of the basis aos In this expression, the Ci are referred to as lcao-Mo coefficients because they tell us how to linearly combine AOs to form the MOs. Because the AOs have various angular shapes(e.g, S, p, or d shapes)and radial extents (i.e, different orbital exponents), the MOs constructed from them can be of different shapes and radial sizes. Let's look at a few examples to see what I mean The first example arises when two h atoms combine to form the H molecule. The valence AOs on each h atom are the ls aOs; they combine to form the two valence mos (o and o*)depicted in Fig. 6.4 2σ Als pBs
25 for a molecule or molecular ion consists of a sum of components involving all of the basis AOs: fj = Sm Cj,m cm . In this expression, the Cj,m are referred to as LCAO-MO coefficients because they tell us how to linearly combine AOs to form the MOs. Because the AOs have various angular shapes (e.g., s, p, or d shapes) and radial extents (i.e., different orbital exponents), the MOs constructed from them can be of different shapes and radial sizes. Let’s look at a few examples to see what I mean. The first example arises when two H atoms combine to form the H2 molecule. The valence AOs on each H atom are the 1s AOs; they combine to form the two valence MOs (s and s*) depicted in Fig. 6.4