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=<oJ-h/2m V-dJ> 2. Coulombic attraction energy with the nucleus of charge Z: <pl-Ze/r lo 3 And Coulomb repulsion energies with all of the n-l other electrons, which are assumed to occupy other atomic orbitals(AOs)denoted ox, with this energy computed as ∑<φr)φs(r”)e2/r-r)|()d(r) The so-called Dirac notation <o(r)x(r)l(e/r-rD,(ox(r)> 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 = <fJ | – h2 /2m Ñ 2 |fJ>; 2. Coulombic attraction energy with the nucleus of charge Z: <fJ | -Ze2 /r |fJ>; 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 <fJ (r) fK(r’) |(e2 /|r-r’|) | fJ (r) fK(r’)>. The so-called Dirac notation <fJ (r) fK(r’) |(e2 /|r-r’|) | fJ (r) fK(r’)> 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 = <fJ | – h2 /2m Ñ 2 |fJ> + <fJ | -Ze2 /|r |fJ> + SK <fJ (r) fK(r’) |(e2 /|r-r’|) | fJ (r) fK(r’)>