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Lecture Handout 2020 Molecular Electronic Structure Quantum Supplementary Hilary Term 2020 Molecular wavefunctions 1. Introduction: Schrödinger equation, Born-Oppenheimer approximation, wavefunctions. 2. The linear combination of atomic orbitals (LCAO) approach to molecular orbitals 3. Pauli principle, Slater determinants + Formulation of the Hamiltonian for H2 + 4. Matrix formulation and the secular determinants for a 1-electron system (H2 ) + 5. Formulation of H2 in the LCAO approach Basis sets 6. Slaters vs Gaussians 7. Hierachy of basis sets Formulation of the Hamiltonian for H2. 8. Coulomb integrals 9. Triplet state of H2 and the exchange integral. 10. Expansion of the Coulomb and exchange integrals in an atomic basis. Self-consistent fields and Hartree-Fock theory 11. Hartree-Fock and Roothaan equations 12. An application:the origin of effective nuclear charge Beyond the HF approximation 13. Configuration interaction 14. Density functional theory Semi-empirical methods 15. Hückel and Extended Hückel theory Bibliography 1. Richards and Cooper, Ab Initio Molecular Orbital Calculations for Chemists. 2. Atkins and Friedman, MQM, 5th ed 3. Lowe, Quantum Chemistry, 2nd ed 3. McQuarrie, Quantum Chemistry, 2nd ed 1 Some basics: In studies of molecular electronic structure, we seek solutions of the time-independent Schrödinger equation, ℏ − ∇2휑 + 푉휑 = 퐸휑 2푚 where m is the mass of an electron, V is the potential and 2 is the Laplacian operator 휕2 휕2 휕2 ∇2= + + 휕푥2 휕푦2 휕푧2 The wavefunction, , must be single-valued, finite and continuous. Born-Oppenheimer approximation The above equation is derived from the complete (electronic + nuclear) Schrödinger equation by regarding the nuclei as being fixed in space, and so the total wavefunction factorises as R,()() r r R The electronic Hamiltonian, Hˆ , for a system with n electrons and N nuclei is then given by 푛 푛 푁 푛 푛 푁 푁 ℏ2 푒2 푍 푒2 1 푒2 1 ̂ 2 푎 퐻 = − ∑ ∇푖 − ∑ ∑ + ∑ ∑ + ∑ ∑ = 퐸휑 2푚 4휋휀0 푟푖푎 4휋휀0 푟푖푗 4휋휀0 푅푎푏 푖 푖 푎 푖 푗>푖 푎 푏>푎 The final term is independent of the position of the electrons, and adds a constant contribution to the energy (at fixed distance) This defines the potential energy curve: the electronic energy which depends parametrically on the nuclear coordinates. Linear Combination of Atomic Orbitals (LCAO) approximation Molecular orbitals are typically expanded as linear combinations of their atomic components: caa a 2 Slater determinants (see Valence Lecture 3, pp 12) The Pauli principle: The total wavefunction must be antisymmetric under the exchange of identical particles. Example: For He, the 2-electron wavefunction 1ss (1)1 (2)is not antisymmetric because 1s (1)1 s (2) 1 s (2)1 s (1) (note 1ss (1)1 (2) is shorthand for 1ss (1)1 (2) (1) (2) ) But the linear combination 1 1s (1)1 s (2) 1 s (2)1 s (1) is antisymmetric because 2 11 1s (2)1 s (1) 1 s (1)1 s (2) 1 s (1)1 s (2) 1 s (2)1 s (1) 22 Example: Li 1s (1)1 s (2)2 s (3) is not antisymmetric. 1 1s (1)1 s (2)2 s (3) 1 s (2)1 s (1)2 s (3) 2 is antisymmetric wrt exchange of electrons 1 and 2, but not 1 and 3 or 2 and 3. 1 1s (1)1 s (2)2 s (3) 1 s (1)1 s (3)2 s (2) 1 s (2)1 s (1)2 s (3) 6 1s (2)1 s (3)2 s (1) 1 s (3)1 s (1)2 s (2) 1 s (3)1 s (2)2 s (1) is antisymmetric wrt exchange of all three. The antisymmetric wavefunction can be written as a Slater determinant 111ss (1) 1 (1) He: 1s (1)1 s (2) 1 s (2)1 s (1) 221ss (2) 1 (2) 1s (1) 1 s (1) 2 s (1) 1 Li: 1(2)s 1 s (2) 2(2) s 6 1(3)s 1 s (3) 2(3) s He (triplet excited state, 12ss11): 111ss (1) 2 (1) 1s (1)2 s (2) 1 s (2)2 s (1) 221ss (2) 2 (2) Note using a single Slater determinant to represent the wavefunction is a convenient way to ensure that it conforms to the Pauli Principle, but it is by no means the only way (any linear combination of Slater determinants will also do the job, see configuration interaction) 3 + H2 : linear variation theorem (n.b. only 1 electron, so Slater “determinant” is just a 1 x 1 determinant: ) trialc n n c a 11 sa c b sb n ˆ ˆ H ca1 sa c b 1 sb H c a 1 sa c b 1 sb Etrial ca1 sa c b 1 sb c a 1 sa c b 1 sb Following the process set out in Prof Brouard’s ‘Valence’ lecture 4 (handout pp 18-23) or the equivalent derivation given in Appendix 1, minimisation of the trial wavefunction gives the following secular determinant: H ES H ES detH ES aa aa ab ab 0 Hab ES ab H bb ES bb with eigenvalues that we can formulate in terms of matrix elements HH aa ab ˆ E and HHaa 11 sa sa etc. 1 Sab (but note we have not yet defined Hˆ !) and eigenfunctions that we can anticipate from symmetry considerations: 1 1 g/ u 1 sa 1 sb 2(1 Sab ) To make further progress we need to convert the abstract “ Haa ”, “ Hab ” etc. into hard numbers. ℏ2 ℏ2 푒2 1 푒2 1 푒2 1 퐻̂ = − ∇2 + 푉 = − ∇2 − ( ) − ( ) + ( ) 2푚 2푚 4휋휀0 푟푎 4휋휀0 푟푏 4휋휀0 푅푎푏 ℏ2 푗 푗 푗 = − ∇2 − 0 − 0 + 0 2푚 푟푎 푟푏 푅푎푏 푒2 푗0 = 4휋휀0 4 푗0 is the repulsion between two nuclei – it is independent of the electron coordinates, so 푅푎푏 adds a constant term to the total energy. We will neglect this term in the following derivations (it can be trivially added in at the end of any energy calculation). A note on units: The Hamiltonian ℏ2 ℏ2 푒2 1 푒2 1 푒2 1 퐻̂ = − ∇2 + 푉 = − ∇2 − ( ) − ( ) + ( ) 2푚 2푚 4휋휀0 푟푎 4휋휀0 푟푏 4휋휀0 푅푎푏 is given in SI units (distance in metres, mass in kg, charge in coulombs, energy in Joules). It is also often expressed in atomic units (distance in multiples of a0 [the Bohr radius], mass in multiples of me, charge in multiples of e, permittivity in multiples of 40, energy in multiples of Hartrees (atomic units). -10 -19 -31 a0=0.529 x 10 m = 0.529 Å e = 1.6022 x 10 C me = 9.1095 x 10 kg 2 ℏ -18 1 hartree = 1 au = 2 = 4.3598 x 10 J = 27.21 eV 푚푒푎0 1 1 1 1 In which case the Hamiltonian simplifies to 퐻̂ = − ∇2 − − + 2 푟푎 푟푏 푅푎푏 See also ‘Valence’ lecture 2, P 8: 1 1 1 Coulomb integral: HH ˆ 2 aa a a a2 rr a ab We will see how to compute these integrals shortly, but for now just note that 112 a a E1 s , the energy of an electron in a hydrogen atom. 2 rA 1 E11s a a E s j' rb 1 2 j' aa= Coulomb attraction between charge density a and nucleus b rb 5 ˆ 12 1 1 Likewise, the resonance integral: HHab a b a b 2 rrab 112 but b E1 s b 2 rb 1 a bE11 s a b SE s k' ra 1 k' ab= interaction between overlap charge density aband nucleus a ra Sub into solutions of secular equation: E11ss j'' SE k jk'' jk'' E EE11 s or EE11 s 1gu /1 11SS g 1 S u 1 S Basis functions In order to evaluate E or E , we need to be able to calculate the various integrals 1 g 1 u involved in the above expressions. These include the overlap integrals, , and the 1 components of H , which are the kinetic energy, 2 , and the electron-nucleus ij 2 1 interaction, . When we get to H2, we will find that we also need to calculate rA 1 electron-electron repulsion integrals of the type (1) (2) (1) (2) r12 To calculate these numbers we need to choose a mathematical representation of the atomic orbitals, . For discrete molecules, there are 2 plausible choices, Slater-type orbitals and Gaussian-type orbitals. For solid-state (periodic) calculations, there is a third choice, a basis set of plane waves, but we will not be concerned with that here. 6 푛−1 −휁푟 Slater-type orbitals (STOs): 푆푇푂 = 푁1푟 푒 푌푙푚(휃, 휙) 푁1 is the normalising constant. 푌푙푚(휃, 휙) is a spherical harmonic, controlling angular dependence 휁 is a measure of how contracted the function is (larger 휁 implies more contracted). In very simple (minimal) basis sets, 휁 is related to Slater’s effective nuclear charge, which controls radial behaviour, and can be calculated using Slater’s simple empirical rules. These replicate the behaviour of real atomic orbitals, so are the most intuitive option. 푖 푗 푘 −훼푟2 푖 푗 푘 −훼(푥2+푦2+푧2) Gaussian-type orbitals (GTOs): 퐺푇푂 = 푁1푥 푦 푧 푒 = 푁1푥 푦 푧 푒 Angular properties are specified by i, j, k: (0,0,0) = s; (1,0,0) = px; (1,1,0) = dxy etc etc. 2 The major difference is that the exponential decay is 푒−훼푟 and not 푒−훼푟, which means that the orbital decays too quickly at large r, and fails to reproduce the cusp at the nucleus found in real atomic orbitals. Thus they are less accurate representations of a real atomic orbital than Slater functions.
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