Chapter 6. Electronic Structures Electrons are the "glue" that holds the nuclei together in the chemical bonds of molecules and ions. Ofcourse, it is the nmuclei's positive charges that bind the electrons to the nuclei. The competitions among Coulomb repulsions and attractions as well as the existence of non-zero electronic and nuclear kinetic energies make the treatment of the full electronic-nuclear Schrodinger equation an extremely difficult problem. Electronic structure theory deals with the quantum states of the electrons, usually within the born- Oppenheimer approximation(.e, with the nuclei held fixed. It also addresses the forces that the electrons presence creates on the nuclei; it is these forces that determine the geometries and energies of various stable structures of the molecule as well as transition states connecting these stable structures. Because there are ground and excited electronic states, each of which has different electronic properties, there are different stable-structure and transition-state geometries for each such electronic state. electroni structure theory deals with all of these states,their muclear structures, and the spectroscopies(e.g, electronic, vibrational, rotational) connecting them . Theoretical treatment of electronic structure: atomic and molecular orbital Theory In Chapter 5s discussion of molecular structure, I introduced you to the strategies that theory uses to interpret experimental data relating to such matters, and how and why
1 Chapter 6. Electronic Structures Electrons are the “glue” that holds the nuclei together in the chemical bonds of molecules and ions. Of course, it is the nuclei’s positive charges that bind the electrons to the nuclei. The competitions among Coulomb repulsions and attractions as well as the existence of non-zero electronic and nuclear kinetic energies make the treatment of the full electronic-nuclear Schrödinger equation an extremely difficult problem. Electronic structure theory deals with the quantum states of the electrons, usually within the BornOppenheimer approximation (i.e., with the nuclei held fixed). It also addresses the forces that the electrons’ presence creates on the nuclei; it is these forces that determine the geometries and energies of various stable structures of the molecule as well as transition states connecting these stable structures. Because there are ground and excited electronic states, each of which has different electronic properties, there are different stable-structure and transition-state geometries for each such electronic state. Electronic structure theory deals with all of these states, their nuclear structures, and the spectroscopies (e.g., electronic, vibrational, rotational) connecting them. I. Theoretical Treatment of Electronic Structure: Atomic and Molecular Orbital Theory In Chapter 5’s discussion of molecular structure, I introduced you to the strategies that theory uses to interpret experimental data relating to such matters, and how and why
theory can also be used to simulate the behavior of molecules In carrying out simulations, the Born-Oppenheimer electronic energy e(r)as a function of the 3N coordinates of the N atoms in the molecule plays a central role. It is on this landscape that one searches for stable isomers and transition states. and it is the second derivative (Hessian) matrix of this function that provides the harmonic vibrational frequencies of such isomers In the present Chapter, I want to provide you with an introduction to the tools that we use to solve the electronic Schrodinger equation to generate e(r) and the electronic wave function Y(rR). In essence, this treatment will focus on orbitals of atoms and molecules and how we obtain and interpret them For an atom, one can approximate the orbitals by using the solutions of the hydrogenic Schrodinger equation discussed in the Background Material. Although such ns are not proper solutions to the actual N-electron Schrodinger equation(believe it or not, no one has ever solved exactly any such equation for n> 1)of any atom, they can be used as perturbation or variational starting-point approximations when one may be satisfied with qualitatively accurate answers. In particular, the solutions of this one- electron Hydrogenic problem form the qualitative basis for much of atomic and molecular orbital theory. As discussed in detail in the background Material, these orbitals are labeled by n, I, and m quantum numbers for the bound states and by I and m quantum numbers and the energy E for the continuum states Much as the particle-in-a-box orbitals are used to qualitatively describe T electrons in conjugated polyenes or electronic bands in solids, these so-called hydrogen like orbitals provide qualitative descriptions of orbitals of atoms with more than a single electron. By introducing the concept of screening as a way to represent the repulsive 2
2 theory can also be used to simulate the behavior of molecules. In carrying out simulations, the Born-Oppenheimer electronic energy E(R) as a function of the 3N coordinates of the N atoms in the molecule plays a central role. It is on this landscape that one searches for stable isomers and transition states, and it is the second derivative (Hessian) matrix of this function that provides the harmonic vibrational frequencies of such isomers. In the present Chapter, I want to provide you with an introduction to the tools that we use to solve the electronic Schrödinger equation to generate E(R) and the electronic wave function Y(r|R). In essence, this treatment will focus on orbitals of atoms and molecules and how we obtain and interpret them. For an atom, one can approximate the orbitals by using the solutions of the hydrogenic Schrödinger equation discussed in the Background Material. Although such functions are not proper solutions to the actual N-electron Schrödinger equation (believe it or not, no one has ever solved exactly any such equation for N > 1) of any atom, they can be used as perturbation or variational starting-point approximations when one may be satisfied with qualitatively accurate answers. In particular, the solutions of this oneelectron Hydrogenic problem form the qualitative basis for much of atomic and molecular orbital theory. As discussed in detail in the Background Material, these orbitals are labeled by n, l, and m quantum numbers for the bound states and by l and m quantum numbers and the energy E for the continuum states. Much as the particle-in-a-box orbitals are used to qualitatively describe pelectrons in conjugated polyenes or electronic bands in solids, these so-called hydrogenlike orbitals provide qualitative descriptions of orbitals of atoms with more than a single electron. By introducing the concept of screening as a way to represent the repulsive
interactions among the electrons of an atom, an effective nuclear charge Zer can be used in place of z in the hydrogenic vnlm and En. formulas of the Background Material to generate approximate atomic orbitals to be filled by electrons in a many-electron atom For example, in the crudest approximation of a carbon atom, the two ls electrons experience the full nuclear attraction so Zeff=6 for them, whereas the 2s and 2p electrons are screened by the two ls electrons, so Zer= 4 for them. Within this approximation, one then occupies two ls orbitals with Z=6, two 2s orbitals with Z-4 and two 2p orbitals with Z=4 in forming the full six-electron product wave function of the lowest-energy state of carbon ,2,…,6)=v1so(1)v1s(2)v20(3)…v1pof阝 However, such approximate orbitals are not sufficiently accurate to be of use in quantitative simulations of atomic and molecular structure. In particular, their energies do not properly follow the trends in atomic orbital(AO)energies that are taught in introductory chemistry classes and that are shown pictorially in Fig 6.1
3 interactions among the electrons of an atom, an effective nuclear charge Zeff can be used in place of Z in the hydrogenic yn,l,m and En,l formulas of the Background Material to generate approximate atomic orbitals to be filled by electrons in a many-electron atom. For example, in the crudest approximation of a carbon atom, the two 1s electrons experience the full nuclear attraction so Zeff =6 for them, whereas the 2s and 2p electrons are screened by the two 1s electrons, so Zeff = 4 for them. Within this approximation, one then occupies two 1s orbitals with Z=6, two 2s orbitals with Z=4 and two 2p orbitals with Z=4 in forming the full six-electron product wave function of the lowest-energy state of carbon Y(1, 2, …, 6) = y1s a(1) y1sba(2) y2s a(3) … y1p(0) b(6). However, such approximate orbitals are not sufficiently accurate to be of use in quantitative simulations of atomic and molecular structure. In particular, their energies do not properly follow the trends in atomic orbital (AO) energies that are taught in introductory chemistry classes and that are shown pictorially in Fig. 6.1
75 Figure 6. 1 Energies of Atomic Orbitals as Functions of Nuclear Charge for Neutral Atoms For example, the relative energies of the 3d and 4s orbitals are not adequately described in a model that treats electron repulsion effects in terms of a simple screening factor. So, now it is time to examine how we can move beyond the screening model and take the electron repulsion effects, which cause the inter-electronic couplings that render the Schrodinger equation insoluble, into account in a more reliable manner A. Orbitals I. The Hartree description
4 Figure 6.1 Energies of Atomic Orbitals as Functions of Nuclear Charge for Neutral Atoms For example, the relative energies of the 3d and 4s orbitals are not adequately described in a model that treats electron repulsion effects in terms of a simple screening factor. So, now it is time to examine how we can move beyond the screening model and take the electron repulsion effects, which cause the inter-electronic couplings that render the Schrödinger equation insoluble, into account in a more reliable manner. A. Orbitals 1. The Hartree Description
The energies and wave functions within the most commonly used theories of atomic structure are assumed to arise as solutions of a Schrodinger equation whose hamiltonian h(r) possess three kinds of energies Kinetic energy, whose average value is computed by taking the expectation value of the kinetic energy operator-A/2m V with respect to any particular solution o, (r) to the Schrodinger equation: KE= 2. Coulombic attraction energy with the nucleus of charge Z: is used to represent the six-dimensional Coulomb integral JIx= J%(r) l%k(r)(e2/r-r,)drdr'that describes the Coulomb repulsion between the charge density lo, (rl for the electron in pj and the charge density p(r)l for the electron in x. Of course, the sum over K must be limited to exclude k=J to avoid counting a self-interaction"of the electron in orbital o with itself The total energy a of the orbital o ,, is the sum of the above three contributions E=-h22mV2|>+<-ze/r伸 +Σx<φr)(r)e/rrD)|ψr)(r)
5 The energies and wave functions within the most commonly used theories of atomic structure are assumed to arise as solutions of a Schrödinger equation whose hamiltonian he (r) possess three kinds of energies: 1. Kinetic energy, whose average value is computed by taking the expectation value of the kinetic energy operator – h2 /2m Ñ 2 with respect to any particular solution fJ (r) to the Schrödinger equation: KE = ; 2. Coulombic attraction energy with the nucleus of charge Z: ; 3. And Coulomb repulsion energies with all of the n-1 other electrons, which are assumed to occupy other atomic orbitals (AOs) denoted fK, with this energy computed as SK . The so-called Dirac notation is used to represent the six-dimensional Coulomb integral JJ,K = ò|fJ (r)|2 |fK(r’)|2 (e2 /|r-r’) dr dr’ that describes the Coulomb repulsion between the charge density |fJ (r)|2 for the electron in fJ and the charge density |fK(r’)|2 for the electron in fK. Of course, the sum over K must be limited to exclude K=J to avoid counting a “self-interaction” of the electron in orbital fJ with itself. The total energy eJ of the orbital fJ , is the sum of the above three contributions: eJ = + + SK
This treatment of the electrons and their orbitals is referred to as the hartree-level of theory. As stated above, when screened hydrogenic AOs are used to approximate the o, and ox orbitals, the resultant E, values do not produce accurate predictions. For example, the negative of E, should approximate the ionization energy for removal of an electron from the Ao. Such ionization potentials(IP s)can be measured, and the measured values do not agree well with the theoretical values when a crude screening approximation is made for the ao s 2. The Laco-Expansion To improve upon the use of screened hydrogenic AOs, it is most common to approximate each of the Hartree AOs(oki as a linear combination of so-called basis AOs x 中=Cmx using what is termed the linear-combination-of-atomic-orbitals (LCAO)expansion. In this equation, the expansion coefficients C) are the variables that are to be determined by solving the Schrodinger equation 中=E1中 After substituting the LCAO expansion for o, into this Schrodinger equaiton, multiplying 6
6 This treatment of the electrons and their orbitals is referred to as the Hartree-level of theory. As stated above, when screened hydrogenic AOs are used to approximate the fJ and fK orbitals, the resultant eJ values do not produce accurate predictions. For example, the negative of eJ should approximate the ionization energy for removal of an electron from the AO fJ . Such ionization potentials (IP s) can be measured, and the measured values do not agree well with the theoretical values when a crude screening approximation is made for the AO s. 2. The LACO-Expansion To improve upon the use of screened hydrogenic AOs, it is most common to approximate each of the Hartree AOs {fK} as a linear combination of so-called basis AOs {cm}: fJ = SmCJ,m cm . using what is termed the linear-combination-of-atomic-orbitals (LCAO) expansion. In this equation, the expansion coefficients {CJ,m} are the variables that are to be determined by solving the Schrödinger equation he fJ = eJ fJ . After substituting the LCAO expansion for fJ into this Schrödinger equaiton, multiplying
on the left by one of the basis AOs %y, and then integrating over the coordinates of the electron in o), one obtains ΣCl=E1EC This is a matrix eigenvalue equation in which the E, and (Clu appear as eigenvalues and eigenvectors. The matrices are called the Hamiltonian and overlap matrices, respectively. An explicit expression for the former is obtained by introducing the earlier definition of he =+ +2kCknCKx Cu=E, Eu xl x>Cu are then solved for new(CKy,) coefficients and for the orbital energies(Ex). The new LCAO coefficients
7 on the left by one of the basis AOs cn , and then integrating over the coordinates of the electron in fJ , one obtains Sm CJ,m = eJ Sm CJ,m . This is a matrix eigenvalue equation in which the eJ and {CJ,m} appear as eigenvalues and eigenvectors. The matrices and are called the Hamiltonian and overlap matrices, respectively. An explicit expression for the former is obtained by introducing the earlier definition of he : = + + SK CK,h CK,g . An important thing to notice about the form of the matrix Hartree equations is that to compute the Hamiltonian matrix, one must know the LCAO coefficients {CK,g} of the orbitals which the electrons occupy. On the other hand, these LCAO coefficients are supposed to be found by solving the Hartree matrix eigenvalue equations. This paradox leads to the need to solve these equations iteratively in a so-called self-consistent field (SCF) technique. In the SCF process, one inputs an initial approximation to the {CK,g} coefficients. This then allows one to form the Hamiltonian matrix defined above. The Hartree matrix equations Sm CJ,m = eJ Sm CJ,m are then solved for “new” {CK,g} coefficients and for the orbital energies {eK}. The new LCAO coefficients
of those orbitals that are occupied are then used to form a"new"Hamiltonian matrix after which the Hartree equations are again solved for another generation of LCAO coefficients and orbital energies. This process is continued until the orbital energies and LCAO coefficients obtained in successive iterations do not differ appreciably. Upon such convergence, one says that a self-consistent field has been realized because the CK,i coefficients are used to form a Coulomb field potential that details the electron-electron Interactions 3.Ao Basis sets STOs and gtos As noted above, it is possible to use the screened hydrogenic orbitals as the xu) However, much effort has been expended at developing alternative sets of functions to use as basis orbitals. The result of this effort has been to produce two kinds of functions hat currently are widely used The basis orbitals commonly used in the lCao process fall into two primary Slater-type orbitals(STOs)Xn, 1,m(r, 0. Q)=Nn. 1, m, Y1m(0. )rn-Ie-sr are haracterized by quantum numbers n, I, and m and exponents(which characterize the orbitals radial'size')s The symbol Nn, l, m, denotes the normalization constant 2. Cartesian Gaussian-type orbitals(GTOs)Xab.c(r, 0,D)=Nab.c a xayb ze exp(-ar4) are characterized by quantum numbers a, b, and c, which detail the angular shape and direction of the orbital, and exponents a which govern the radial ' size For both types of AOs, the coordinates r, 0, and o refer to the position of the 8
8 of those orbitals that are occupied are then used to form a “new” Hamiltonian matrix, after which the Hartree equations are again solved for another generation of LCAO coefficients and orbital energies. This process is continued until the orbital energies and LCAO coefficients obtained in successive iterations do not differ appreciably. Upon such convergence, one says that a self-consistent field has been realized because the {CK,g} coefficients are used to form a Coulomb field potential that details the electron-electron interactions. 3. AO Basis Sets a. STOs and GTOs As noted above, it is possible to use the screened hydrogenic orbitals as the {cm}. However, much effort has been expended at developing alternative sets of functions to use as basis orbitals. The result of this effort has been to produce two kinds of functions that currently are widely used. The basis orbitals commonly used in the LCAO process fall into two primary classes: 1. Slater-type orbitals (STOs) cn,l,m (r,q,f) = Nn,l,m,z Yl,m (q,f) rn-1 e-zr are characterized by quantum numbers n, l, and m and exponents (which characterize the orbital’s radial 'size' ) z. The symbol Nn,l,m,z denotes the normalization constant. 2. Cartesian Gaussian-type orbitals (GTOs) ca,b,c (r,q,f) = N'a,b,c,a xa yb zc exp(-ar2), are characterized by quantum numbers a, b, and c, which detail the angular shape and direction of the orbital, and exponents a which govern the radial 'size'. For both types of AOs, the coordinates r, q, and f refer to the position of the
electron relative to a set of axes attached to the nucleus on which the basis orbital is located. Note that Slater-type orbitals(STOS) are similar to hydrogenic orbitals in the region close to the nucleus. Specifically, they have a non-zero slope near the nucleus (i.e d/dr(exp(-Cr))=0=-s). In contrast, GTOS, have zero slope near r=0 because d/dr(exp(-ar))20=0. We say that STOs display a cusp"at r=0 that is characteristic of the hydrogenic solutions, whereas GtOs do not Although STOs have the proper'cusp' behavior near nuclei, they are used primarily for atomic and linear-molecule calculations because the multi-center integrals which arise in polyatomic-molecule calculations(we will discuss these intergrals later in this Chapter) can not efficiently be evalusted when STOs are employed. In contrast, such integrals can routinely be computed when GTOs are used. This fundamental advantage of gtos has lead to the dominance of these functions in molecular quantum chemistry To overcome the primary weakness of GTO functions(i.e, their radial derivatives vanish at the nucleus), it is common to combine two, three or more gtOs, with combination coefficients which are fixed and not treated as lcao parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of radially tight medium, and loose GTOs are multiplied by contraction coefficients and summed to produce a CGto which approximates the proper'cusp' at the nuclear center(although no such combination of GTOs can exactly produce such a cusp because each Gto has zero slope at r=0) Although most calculations on molecules are now performed using Gaussian orbitals, it should be noted that other basis sets can be used as long as they span enough 9
9 electron relative to a set of axes attached to the nucleus on which the basis orbital is located. Note that Slater-type orbitals (STO's) are similar to hydrogenic orbitals in the region close to the nucleus. Specifically, they have a non-zero slope near the nucleus (i.e., d/dr(exp(-zr))r=0 = -z). In contrast, GTOs, have zero slope near r=0 because d/dr(exp(-ar 2 ))r=0 = 0. We say that STOs display a “cusp” at r=0 that is characteristic of the hydrogenic solutions, whereas GTOs do not. Although STOs have the proper 'cusp' behavior near nuclei, they are used primarily for atomic and linear-molecule calculations because the multi-center integrals which arise in polyatomic-molecule calculations (we will discuss these intergrals later in this Chapter) can not efficiently be evalusted when STOs are employed. In contrast, such integrals can routinely be computed when GTOs are used. This fundamental advantage of GTOs has lead to the dominance of these functions in molecular quantum chemistry. To overcome the primary weakness of GTO functions (i.e., their radial derivatives vanish at the nucleus), it is common to combine two, three, or more GTOs, with combination coefficients which are fixed and not treated as LCAO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of radially tight, medium, and loose GTOs are multiplied by contraction coefficients and summed to produce a CGTO which approximates the proper 'cusp' at the nuclear center (although no such combination of GTOs can exactly produce such a cusp because each GTO has zero slope at r = 0). Although most calculations on molecules are now performed using Gaussian orbitals, it should be noted that other basis sets can be used as long as they span enough
of the regions of space(radial and angular where significant electron density resides. In fact, it is possible to use plane wave orbitals of the form x(r, 0, d)=N exp[i(kx r sine cosd +k, r sine sind+k, r cose), where N is a normalization constant and k ky, and k, are quantum numbers detailing the momenta of the orbital along the x, y, and z Cartesian directions. The advantage to using such "simple" orbitals is that the integrals one must perform are much easier to handle with such functions. The disadvantage is that one must use many such functions to accurately describe sharply peaked charge distributions of for example, inner-shell core orbitals Much effort has been devoted to developing and tabulating in widely available locations sets of STO or GtO basis orbitals for main-group elements and transition metals. This ongoing effort is aimed at providing standard basis set libraries which 1. Yield predictable chemical accuracy in the resultant energies 2. Are cost effective to use in practical calculations 3. Are relatively transferable so that a given atom s basis is flexible enough to be used fo that atom in various bonding environments(e.g, hybridization and degree of ionization) b. The fundamental core and valence basis In constructing an atomic orbital basis, one can choose from among several classes of functions. First, the size and nature of the primary core and valence basis must be specified. Within this category, the following choices are common a minimal basis in which the number of cto orbitals is equal to the number of core and valence atomic orbitals in the atom 2. A double-zeta(dz)basis in which twice as many CGTOs are used as there are core
10 of the regions of space (radial and angular) where significant electron density resides. In fact, it is possible to use plane wave orbitals of the form c (r,q,f) = N exp[i(kx r sinq cosf + ky r sinq sinf + kz r cosq)], where N is a normalization constant and kx , ky , and kz are quantum numbers detailing the momenta of the orbital along the x, y, and z Cartesian directions. The advantage to using such “simple” orbitals is that the integrals one must perform are much easier to handle with such functions. The disadvantage is that one must use many such functions to accurately describe sharply peaked charge distributions of, for example, inner-shell core orbitals. Much effort has been devoted to developing and tabulating in widely available locations sets of STO or GTO basis orbitals for main-group elements and transition metals. This ongoing effort is aimed at providing standard basis set libraries which: 1. Yield predictable chemical accuracy in the resultant energies. 2. Are cost effective to use in practical calculations. 3. Are relatively transferable so that a given atom's basis is flexible enough to be used for that atom in various bonding environments (e.g., hybridization and degree of ionization). b. The Fundamental Core and Valence Basis In constructing an atomic orbital basis, one can choose from among several classes of functions. First, the size and nature of the primary core and valence basis must be specified. Within this category, the following choices are common: 1. A minimal basis in which the number of CGTO orbitals is equal to the number of core and valence atomic orbitals in the atom. 2. A double-zeta (DZ) basis in which twice as many CGTOs are used as there are core
