2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements
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1 1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical elements 6 Diatomic molecules 7 Ten-electron systems from the second row 8 More complicated molecules FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- The Schrödinger equation Combining the classical Hamilton function for a free particle with mass m, H = p /m, with the de Broglie relation p = k and the Planck relation E = ε = ω leads to H = E = T = p m = k m = ω, k = k x + k y + k z 0 With eq 19, this relation is readily transferred into an operator identity for a differential equation wave equation?: i m x Ĥ = + m x y + m k x + k y + k z = ω i y + i = i z + = z i m = i 1 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
2 This operator identity can be applied to a suitable mathematical object, eg a complex-valued scalar function ψr, t of position r and time t, to give the time-dependent Schrödinger equation: Ĥψ = i ψ Ĥ i ψ = 0 The function ψr, t is called state function, or wave function As in classical mechanics, where the Hamilton function includes kinetic and potential energies, H = T +V, the Hamilton operator is extended to include parts for both kinetic and potential energy, Ĥ = T + V For application to scalar functions ψr, t we thus have: p p = i 3 T = p m T i = m = m = m 4 V = V r, t V = V r, t 5 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- Gradient operator and Laplace operator in cartesian coordinates: x =, = = = y x + y + z z From eq and its complex conjugate assuming V to be real, T + V i ψ = 0 T + V + i ψ = 0 6 follows ψ T + V i ψ = 0 ψ T + V + i ψ = 0 7 The difference of these two expressions is the terms with V cancel ψ m ψ i ψ ψ ψ m ψ i ψ ψ = 0 8 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
3 which gives, after re-ordering ψ ψ + ψ ψ i m ψ ψ + i m ψ ψ = 0 9 This is the continuity equation known, eg, from electrodynamics ρ + j = 0 30 with real-valued distributions for density ρ and current j of some property defined as ρ = ψ ψ, j = i ψ ψ ψ ψ 31 m These are density and current for the quantity probability M Born, so that units can be assigned as follows in 3-dimensional cases: [ρ] = m 3, [ψ] = m 3/, and also [j x,y,z ] = ms 1 m 3 Normalization of the probability density distribution: V ρ dr = V ψ ψ dr = N < 3 The integration may be extended over the full space R 3, if the integral exists FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- When the Hamilton operator Hamiltonian does not depend on time, ie Ĥ Ĥt, eq has solutions of the form ψr, t = φr exp i E t/ 33 which represent stationary states, provided the spatial part φr satisfies the time-independent Schrödinger equation: Ĥ Ĥφ = E φ E φ = 0 34 The separation constant E is the energy of the stationary state The probability density for stationary states does not depend on time: ρ = ψ r, t ψr, t = φ r φr 35 Solving eq 34 is equivalent to a variation principle, applied to the integral functional of φ H[φ] = φ Ĥφ dr 36 This is an application of a general theorem from the theory of partial differential equations PDE: If the partial differential operator is a sum of terms, where each term acts on a different coordinate or set of coordinates, then there are solutions in product form, and the PDE can be split into lower-dimensional parts FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
4 under the condition that φ remains normalized use Lagrange s method of undetermined multipliers Consider the functional L[φ] = H[φ] λ φ φ dr 1 = φ Ĥφ dr λ φ φ dr 1 37 λ is a yet undetermined Lagrange parameter Condition for stationarity: δl = 0, with δl = δφ Ĥφ dr + φ Ĥδφ dr λ δφ φ dr + φ δφ dr = δφ Ĥ [Ĥ λ φ dr + λ φ ] δφ dr 38 The first integral is the complex conjugate of the second integral if Ĥ = Ĥ Since δl = 0 shall hold for arbitrary variations δφ, it follows: Ĥ λ φ = 0 λ = E 39 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- Together with boundary conditions for φ, the time-independent Schrödinger equation thus constitues an eigenvalue problem, which has to be solved for pairs E, φ of eigenvalues E and eigenfunctions φ Again a situation comparable to eg classical mechanics is found: There exists a variation principle behind the working equations In classical mechanics, the working equations are the Lagrange or Hamilton equations one equation, or a pair of equations, for every degree of freedom, whereas in quantum mechanics, the working equation for stationary states of the system under study is the timeindependent Schrödinger equation FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
5 The free particle This is the quantum mechanical analogue of the situation considered with Newton s first law of motion: The potential energy V is constant may be set to zero, and the particle with mass m has only kinetic energy, so that V = 0, Ĥ = T + V = T = m 40 It follows immediately from our derivation of the time-dependent Schrödinger equation, that plane waves are acceptable solutions k ψr, t = C exp [ ik r ω t ], E = ω = m 0 41 which are also eigenfunctions to the linear momentum vector operator p with eigenvalue vector k: p ψ = i ψ = k ψ 4 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- The probability density is constant, ρ = ψ ψ = C C = C, everywhere and at every time This is OK for a plane wave The most general solution is obtained by continuous superposition note the relation to Fourier transformation ψr, t = Ck e ik r ω t k dk, ωk = 43 m since integration in k-space can be interchanged with differentiation with respect to ordinary space coordinates or time A suitable choice for Ck leads to a valid representation of a free particle a well located object, which is also known as a wave packet FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
6 The particle in a box This is the quantum mechanical analogue of a classical particle with mass m confined to a 1-, - or 3-dimensional region of space: The potential energy V is constant may be set to zero within this region, but infinitely high outside For a 3-dimensional rectangular box with lengths L x, L y, and L z : 0 0 x L x, 0 y L y, 0 z L z V = 44 elsewhere Inside the box, the time-dependent Schrödinger equation has the form Ĥ i ψ = m i ψ = 0 45 which has stationary solutions of the form ψr, t = φr e i E t/, φr = Xx Y y Zz, E = E x + E y + E z 46 and φr has to vanish at the boundaries of the box FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- The problem can be reduced to one-dimensional problems of the following form: d mdx E x Xx = 0 d X dx + k x Xx = 0 k x = m E x 47 with boundary conditions X0 = 0 and XL x = 0 The general solution of the ordinary differential equation 47 is Xx = c 1 e i k x x + c e i k x x = A cos kx x + B sin k x x 48 with A = c 1 + c, B = ic 1 c The boundary conditions lead to A = 0 from X0 = 0 and k x L x = n π with n Z from XL x = 0, so that the acceptable solutions are X n x = B sin k x x, with n > 0 k x = n π, E x = k x L x m = h n 49 8 m L x n = 0 gives X0 x 0, while n < 0 gives the same energy and probability density, since X n = X n FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
7 The coefficient B is determined by normalization of the probability density: 1 = Lx 0 X x Xx dx = B L x sin k x x dx B = 0 L x 50 The boundary conditions lead to quantization This is a general result, as well as the fact, that the number of zeros nodes of the solutions X n x increases with the energy compare with a string The energies eigenvalues grow quadratically as a function of n, E x n n, and the lowest energy is higher than the potential energy minimum, E x n = 1 > 0 The particle in a 1-dimensional box has at least this energy, even at temperature T = 0 K zero-point energy FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- State functions X n x and probability density distributions [X n x] X n x for n = 1,, 3 for a particle in a 1-dimensional box E h /8mL x x/l x n = 3 V x n = n = 1 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
8 Extension to the general case of a 3-dimensional rectangular box is straightforward: 8 φ nkl r = sin nπ xlx sin kπ yly sin lπ zlz 51 L x L y L z E nkl = E x + E y + E z = h 8m n L x + k L y + l L z The ground state ie the state with lowest possible energy: 8 φ 111 r = sin π xlx sin π yly sin π zlz L x L y L z E 111 = h 1 8m L + 1 x L + 1 y L z Whenever there is a rational relation between any two or all three squared lengths L x, L y, and L z, degeneracies occur, ie several state functions belong to the same energy eigenvalue E g in the case of a cubic box L x = L y = L z = L: E nnn is non-degenerate, E nnl n l is triply degenerate, and E nkl n, k, l all different is sixfold degenerate FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- Energy levels E nkl and degrees of degeneracy g for a particle in a 3-dimensional cubic box E nkl = h n + k + l 8mL n k l n + k + l g E h /8mL E 3 3 E 13 6 E 1 E E 1 3 E 11 3 E FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
9 Special functions The gamma function: Γa = 0 ta 1 e t dt a > 0 Γa + 1 = a Γa Γ1 + a Γ1 a = π a sin π a Γn + 1 = n! n N Γ1/ = π The Pochhammer symbol: Γa + k 1 k = 0 a k = = Γa aa + 1 a + k 1 k > 0 1 n = n! a k = 1 k a k + 1 k L Pochhammer FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- The confluent hypergeometric function Kummer function: 1F 1 a; c; x = k=0 a k x k c k k! = 1 + a aa + 1 x x + c cc + 1! + is a solution of the Kummer differential equation x y + c x y a y = 0 The Gauß hypergeometric function: F 1 a, b; c; x = k=0 = 1 + ab c a k b k c k x k k! x + aa + 1bb + 1 cc + 1 is a solution of the Gauß differential equation x! + x 1 x y + [ c a + b + 1 x ] y ab y = 0 E E Kummer FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
10 The harmonic oscillator The 1-dimensional harmonic oscillator is the quantum mechanical analogue of a classical particle with mass m, moving in a parabolic potential or, equivalently, moving under a restoring force given by Hooke s law with force constant K V x = 1 K x F x = dv dx = K x 55 Classically, the particle can oscillate with arbitrary energy, the oscillating frequency ν being related to the force constant K through ω = πν = K m K = m ω 56 Quantum mechanically, the Schrödinger equation, corresponding to the potential energy V x given above, has to be solved FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- Stationary states φx, with x, are solutions of with d m dx + 1 K x E φx = 0 57 d dx + k λ x φx = 0 58 k = m E, λ = m ω The variable transformation ξ = λ x gives d dξ + ε ξ φξ = 0, ε = k λ = E ω 59 Physically acceptable ie normalizable solutions of this ordinary differential equation exist only for ε n = n + 1, n = 0, 1,, E n = n + 1 ω 60 Quantization, due to boundary conditions φξ 0 for ξ FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
11 These solutions are, in explicit form, φ n x = N n H n ξ exp 1 ξ, ξ = m ω K x, ω = m 61 1 m ω 1/ N n = n, E n = n + 1 ω, n = 0, 1,, n! π As the mass m increases, the energy spacing E n+1 E n = ω decreases Thus, the classical behaviour is recovered in the limit m and also for h 0 The Hermite polynomials H n ξ can be expressed in terms of confluent hypergeometric functions, and obey a recurrence relation: H k ξ = 1 kk! 1F 1 k; 1/; ξ k! 6 H k+1 ξ= 1 kk + 1! ξ 1 F 1 k; 3/; ξ k! 63 H n+1 ξ = ξh n ξ nh n 1 ξ n 1 64 Ch Hermite FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- The first Hermite polynomials n H n ξ ξ 4ξ 3 8ξ 3 1ξ 4 16ξ 4 48ξ ξ 5 160ξ ξ FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
12 The state function for the ground state n = 0 is a Gauß func tion φ0 x = exp ξ // 4 π, with energy E0 = ~ ω/ zero-point energy FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- State functions φnx and probability density distributions [φn x] φnx for n = 0, 1,, 3, 4, and 5 for the 1-dimensional harmonic oscillator V x φ5x, E5 = 11~ ω/ n=5 φ4x, E4 = 9~ ω/ n=4 φ3x, E3 = 7~ ω/ n=3 φ x, E = 5~ ω/ n= φ1 x, E1 = 3~ ω/ n=1 φ0x, E0 = ~ ω/ n=0 5 5 x FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
13 The general molecule After these one-particle examples the step to the case of an arbitrary molecule can be done quite easily, if we restrict ourselves to the nonrelativistic treatment, and omit external electric or magnetic fields Hamilton operator for an arbitrary molecule, built from N nuclei with charges Z K and masses M K, and n electrons definition of the system under study, in space-fixed coordinates: Ĥ = T + V 65 Kinetic energy operator based on single-particle kinetic energy: T = T n + T e, T n = N K=1 M K K, T e = m e n i i=1 66 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- Potential energy operator based on the Coulomb potential energy: V = V nn + V en + V ee 67 V nn = + e κ 0 V en = e κ 0 N 1 N K=1 L=K+1 n N i=1 K=1 V ee = + e n 1 κ 0 n i=1 j=i+1 Z K Z L R KL 68 Z K r ik 69 1 r ij 70 One can try to separate motions related to the various degrees of freedom, as in classical mechanics This can be done exactly for the overall translation of the center of total molecular mass, but a separation of the internal nuclear motions molecular rotation and nuclear vibration from the electronic motion is possible only in an approximate way FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
14 Switching to molecule-fixed coordinates, and making a sequence of well-defined steps adiabatic ansatz, adiabatic approximation, Born- Oppenheimer approximation leads to the so called electronic Schrödinger equation for space-fixed or clamped nuclei : Ĥ el Ψ k = E k Ψ k, k = 0, 1,, 71 The electronic Hamiltonian can be split into zero-, one- and twoelectron parts: Ĥ 0 = e κ 0 Ĥ 1 = Ĥ el = Ĥ0 + Ĥ1 + Ĥ 7 N 1 N K=1 L=K+1 n i=1 Ĥ = e n 1 κ 0 Z K Z L R KL i e m e κ 0 n 1 i=1 r j=i+1 ij N Z K K=1 r ik FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9- The zero-electron part Ĥ0 contributes only an additive constant to the total energy and is thus sometimes excluded from Ĥel The electronic energies E k eigenvalues and the corresponding electronic state functions Ψ k eigenfunctions depend parametrically on the internal coordinates of the nuclear configuration In addition to this parametrical dependence on nuclear configuration, the state functions are functions of the space and spin coordinates of all n electrons ie all those coordinates which have to or may appear due to the form of the Hamilton operator Ĥel : Ψ k = Ψ k x 1, x,, x n, 73 x = r, σ combines space and spin coordinates the latter can take only two values: σ = + 1 for or α spin, σ = 1 for or β spin The electronic energy E k, also known as potential energy hypersurface PES, is a scalar function in an f-dimensional space, where f depends FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
15 on the number of nuclei N: non-linear molecule: f = 3N 6, linear molecule: f = 3N 5 For a diatomic molecule N = f = 1, and the electronic energy simply reduces to potential energy curves E k R, where R denotes the internuclear distance All quantum chemical program codes only determine approximate solutions E k, Ψ k to eq 71 A quantum mechanical treatment of nuclear motion, where the PES E k would appear as a part of an effective potential energy operator, is not attempted Most of present-day quantum mechanical methods only differ 1 in the degree of sophistication applied to approximate the state functions Ψ k and/or the presence or absence of further approximations made with respect to the electronic Hamiltonian Ĥel and the evaluation of the energies E k this statement excludes methods based on density functional theory, DFT FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 9-
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