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Solution of the Schrodinger Equation for multi electron atomic systems cannot be done with great precision. It is because of the repulsion energy terms of the potential energy of such systems cannot be handled mathematically accurately, analytically. Many Electron Atoms Chapter 21 Approximate (numerical) methods however handle such systems numerically. Such numerical methods has allowed to consider the complications brought about in multi-electron systems; such as indistinguishability of electrons, electron spin, interaction between orbital & spin magnetic moments. Time-independent Schrödinger Equation for H atom: ∇= Wavefunction Separation of variables KE PE ∇ el2 = ∂ ∂ ∂ + + ∂x ∂y ∂z and ∇ 2 = ∂2 ∂2 ∂2 + + ∂x 2 ∂y 2 ∂z 2 KE of the electron ∂2 ∂2 ∂2 + 2+ 2 2 ∂x ∂y ∂z H atom PE ∇ el2 KE H atom: PE r PE PE term is spherically symmetric; 1/r r H atom He Atom – two electron system Orbital Approximation: ∇ el2 ,1 + ∇el2 ,2 As the number of electrons the ri’s increase and so are the number of rij’s. Vel ,1 + Vel ,2 = r12 V1,2 SE for He atom: He atom V1,2 Insoluble analytically. They are not spherically symmetrical and appears in the repulsive potential energy terms between electrons. Such repulsive terms with rij’s cannot be handled mathematically in an analytical manner, only numerically. Therefore a wave function relating all coordinates of n-electrons as a single function would be impossible. el 2 SE ⇒ H atom r el 1 - distribution If there is a way to treat all the electrons as independent, with no rij appearing between electrons, the solution of SE would yield a wave functions for each electron. r2 r2 r1 Now, the effect of electron 1 (el1) on el2 would be a repulsion that minimize the attractive force on the el 2 by the nucleus. The repulsion would be spherically symmetric!. He atom In the orbital approximation, the many (n) electron wave function is expressed as a product of functions where each function depend only on the coordinates of one electron. The functional form of each one electron orbitals φi (ri) are similar to the H atom orbitals ψ nlml(r,θ,φ); like 1s, 2s, 2p etc., with associated energies. Place the second electron in the spherically symmetric potential field due to nucleus and the spherically symmetric electron distribution from el 1. H like 1s of He atom φi (ri) = ‘orbital’ This approach makes solution of an n-electron wave function from SE, to solving n, one-electron SE equations. Solution of which would yield n, φ’s (atomic orbitals) H like 1s of n electron atom 1 Z π a0 1 ζ π a0 3/ 2 − e 3/ 2 − e Zr a0 ζ<Z ζr a0 ζ = effective nuclear charge (< Z) felt by the electron. el 2 Z ζ<Z el 1 distribution to use for Veff calculation. The effect of el 1 on el 2 is like smearing the electrons e1 uniformly around the atom. It makes the effective nuclear charge < nuclear charge, Z. It would be the same for el 1 due to el 2. el 1 el 2 distribution to use for Veff calculation. Finding ζ is an iterative process iteration done until it is self consistent. Electron Spin z In multi-electron systems the electron spin plays an important role in formulating the SE. s Electron spin with has two intrinsic states, two z-components of spin momentum, as shown below, 1 1 + ℏ or - ℏ . 2 2 1 sz = ± ℏ . 2 The intrinsic electron spin is a vector (angular momentum) s. with q.n. ms = +1/2 and ms = -1/2 and the respective spin wavefunctions α and β (α & β are orthogonal). sz sz s One electron wave function (a wave function associated with one electron) will then have a set of four quantum numbers. n, l , ml and ms i.e.ψ n ,l ,ml ,ms ( r ,θ , φ ). ms = + 1 2 ms = − 1 2 Spin function is not dependent on r , θ or φ . and is intrinsic. orthogonality Spin variable 1sα So the wave function of which the space part is; 1sβ spatial 1s wave functions redefined – spin orbital. spin energy The wave function describing an electron in an atomic orbital has two parts, space and spin parts e.g, 1s(1)α(1), 1s(1)β(1); an electron ‘labeled’ 1 in 1s space orbital with α spin and electron ‘labeled’ 1 in 1s space orbital with β spin. must enjoin with its spin part to form the total wave function. For He a two electron atom ψ(r1, r2)= ; 1sα 1sβ 1s* 1s The indistinguishability of electrons makes identifying electrons in an atomic system with labels like 1, 2, .. etc meaningless; ψ (1, 2) = φ1 (1)α (1)φ2 ( 2)β ( 2) ψ ( 2,1) = φ1 ( 2)α ( 2)φ2 (1)β (1) symmetric interchange of elecrtrons 1sα 1sα and also if ψ is expressed as ψ ( 2,1) = −φ1 ( 2)α ( 2)φ2 (1)β (1) antisymmetric is acceptable because ψ 2 would be the same 1s* 1s in both cases. Postulate 6 deals with the indistinguishable character of the electrons in a multi-electron system after Pauli. φ1s (1)α (1)φ1s ( 2)β ( 2) Postulate 6 requires the wavefunction be anti-symmetric (changes sign) with respect to exchange of any two ‘electron labels’ φ1s ( 2)α ( 2)φ1s (1)β (1) Linear combination of above - a better representation. A [φ1s (1)α (1)φ1s ( 2)β ( 2) + φ1s ( 2)α ( 2)φ1s (1)β (1)] But Not anti-symmetric. Anti-symmetric!! Slater Determinant (total wave function – ground state) φ (1)α (1) ψ (1, 2) = A 1s φ1s ( 2)α ( 2) φ1s (1)β(1) φ1s (2)β(2) 1 φ1s (1)α (1) 2 φ1s ( 2)α ( 2) φ1s (1)β(1) φ1s (2)β(2) ψ (1, 2) = total wave function Anti symmetric character is in engrained in the spin part of the total wave function. Each column same spin orbital Each row same electron label n = # electrons 1s12s1 m = n/2 (n even) or =(n/2 +1/2) (n odd) (total wave function – excited states He) 3Li (ground state) If the three electrons are placed in the same orbital the total wave function would be; φ1s (1)α (1) ψ (1, 2, 3) = φ1s ( 2)α ( 2) 3 φ1s (3)α (3) 1 1 2 3 4 φ1s (1)β(1) φ1s (2)β(2) φ1s (3)β(3) φ1s (1)α (1) φ1s (2)α (2) φ1s (3)α (3) 1 ψ (1, 2) = A φ1s (1)α (1) φ1s ( 2)α ( 2) φ2 s (1)β(1) φ2 s (2)β(2) 2 ψ (1, 2) = A φ1s (1)α (1) φ1s ( 2)α ( 2) φ2 s (1)α (1) φ2 s (2)α ( 2) ψ (1, 2, 3) = 0 No wave function at all !! 3 ψ (1, 2) = A φ1s (1)β (1) φ1s ( 2)β ( 2) φ2 s (1)β(1) φ2 s (2)β(2) 4 ψ (1, 2) = A φ1s (1) β (1) φ1s ( 2)β ( 2) φ2 s (1)α (1) φ2 s (2)α (2) i.e. three electrons (or more) cannot occupy the same space orbital, max number of electrons in a space orbital is two; Pauli Exclusion Principle. The space part of the wave function is the hard to determine in any multi electron system. Approximate Wave Functions and the Variation Method They are H like functions but are not the same. For many situations it is not practical to obtain a wave function by the solution of the wave equation that describes the system. H like functions on different atoms are not easy to integrate as well. Yet, it is possible to perform calculations using the Variation Theorem. Variation Theorem Energy associated with a wave function ψ 0 (true, exact wave function) is calculated using; E0 = ∫ψ Hψ dτ ∫ψ ψ dτ * 0 0 * 0 0 For any other (trial) wave function ψ which is an approximation (a function constructed to emulate the state ψ 0) the calculated energy would be; E= ∫ψ ∫ψ * trial Hψ trial dτ ψ trial dτ * trial ≥ E0 Improving the constructed trial function ψ trial would improve the calculated E and in it’s limit will be equal to E0. The trial functions usually contain parameters and finding the best parameters is the objective of the variation method. Variation theorem provides the basis for the variation method, which states any trial function other than the true function would give an energy E that is higher than the true energy E0. A trial function will be constructed with adjustable parameters so that it could be varied to improve the ‘trial function’. The best parameterized parameters (optimal values) would give the lowest (minimum) energy corresponding to the trial function, closer to E0. In the case of atoms the functions of the form of the H atom solutions is a good starting point. Set α = 0 and calculate; E1 = 0.125 h2 ↔ψ 0 ma 2 E= ∫ψ ∫ψ * trial Hψ trial dτ ψ trial dτ * trial Illustration: Particle in a Box Fact: H =− ℏ2 d 2 2m dx 2 E1 = h2 h2 = 0 . 125 8ma 2 ma 2 An acceptable trial wave function for n=1 state is; Changing α would change the energy estimated. Variation Method The best value for α gives the lowest possible energy (minimizes energy) for the trial function. Variation method allows us to find the ‘optimized value for the parameter’ - parameterization. a. Find the expression for energy in terms of the parameter (s) b. Partial differentiation w.r.t each parameter and set the resulting expression to zero. c. Solve for parameters. Illustration: Particle in a Box - continuation Differentiating E w. r. t. α and setting it to zero gives α = -5.74 and -0.345; latter gives the minimum energy. E1 = 0.127 h2 ma 2 E1 = 0.125 h2 ↔ψ 0 ma 2 Completing the integration gives an expression for E. Self-consistent field theory α=0 An important unsolved problem in QM - how to deal with indistinguishable, interacting particles. The key to success of the Variation Method is the selection of a trial function. The basic problem is that if particles interact, that interaction must represented in the Hamiltonian. α= -0.345 So until we know where the particles are, we can’t write down the Hamiltonian, and until we know the Hamiltonian, we can’t tell where the particles are. Hartree-Fock theory Hartree: To solve the Schrödinger equation for an electron moving in the potential of the nucleus and all the other electrons, start with a guess for the trial electron charge density. Solve n/2 one-particle Schrödinger equations to obtain n electron wave functions. Then construct the potential for each wave function from that of the nucleus and that of all the other electrons, symmetrize it, and solve the n/2 Schrödinger equations again. Fock: Improved on Hartree’s method by using the properly anti-symmetrised wavefunction (Slater determinant) instead of simple one-electron wavefunctions. This accounts for the exchange interaction. The methodology is used (HF-SCF) for molecules. This is a variational method, so wherever we refer to wavefunctions, we assume that they are expanded in some appropriate basis set. Written as an algorithm. HF-SCF Method el 2 el 1 distribution to use for Veff calculation. Guess ψ Improved old ψ improved ψ el 1 el 2 distribution to use for Veff calculation. ψ !! Solving SE for many electron systems. Use orbital approximation and Pauli exclusion principle. Every electron feels the spatially averaged electron charge distribution by the rest of the (n-1) electrons. Antisymmetric wave function (as Slater determinant) No rij terms, so spherically symmetric potential exist; angular part of the orbitals are same as that of H atom. Every φj is a modified H atom orbital. Radial part φj(r) varies from case to case and the variation method allows us to find the optimum parameters of the radial part of wave function. The functions for φj(r) Each φj(r) of the determinant is constructed as a linear combination of some well behaved (easily mathematically handled ) basis functions fi(r). e.g. acceptable φ(r) for 1s of H 1s Variational parameters m=3 Variational parameter suitable Less suitable best fit 1s sum of 3 Gaussians e.g. acceptable φj(r) for 2p. 2p Variational parameters f i ( r ) = N i r exp( −ζ i r / a0 ) m=4 Hartree Fock radial functions can be used to obtain the radial probability for multi-electron atoms. ni = # electrons in the shell. P(r) In multi-electron atoms the energy of an orbital depends on (n + l), for same (n + l) values, the higher n value takes precedence, unlike in H atom (n only)