Quantum Chemistry A Concise Reference of Formulae, Concepts & Data
Dr. Wissam Helal, Department of Chemistry, The University of Jordan, [email protected]
October 10, 2020
About this document
This document is intended to be a concise desk reference of quantum chemistry to be used by both undergraduate and graduate students. It contains three main parts: basic formulas, concepts, and symbols of quantum chemistry (sections 1 – 6); a short review of relevant mathematics and classical physics (sections 7 & 8); and some miscellaneous other useful data (sections 9 & 10). It should be emphasized that this document is not intended to be a summary of any quantum chemistry course, but rather, a useful handout to be used when working out problems. Moreover, this short refernce is far to be comprehensive and do not cover all aspects of the field.
Contents 7 Mathematics 11 7.1 Algebra, Geometry & Trigonometry ...... 11 1 Quantum Mechanics 2 7.2 Derivatives & Integrals ...... 12 1.1 Formulae of Quantum Mechanics ...... 2 7.3 Table of Integrals ...... 12 1.2 Postulates of Quantum Mechanics ...... 3 7.4 Power Series ...... 13 1.3 Dirac Notation (Bracket Notation) ...... 3 7.5 Spherical Polar Coordinates ...... 13 1.4 Theorems of Quantum Mechanics ...... 3 7.6 Complex Numbers ...... 13 7.7 Vectors ...... 14 2 Systems with Exact Solutions 4 7.8 Determinants ...... 15 2.1 The Particle in a Box ...... 4 7.9 Simultaneous Linear Equations ...... 15 2.2 The Particle in a Ring ...... 4 7.10 Matrices ...... 15 2.3 The Harmonic Oscillator ...... 4 7.11 Eigenvalues and Eigenvectors ...... 16 2.4 Angular Momentum ...... 5 2.5 The Rigid Rotor ...... 5 8 Classical Physics 17 2.6 The Hydrogen (Hydrogenlike) Atom ...... 5 8.1 Classical Mechanics ...... 17 8.2 The Classical Wave Equation ...... 18 3 Approximation Methods 7 8.3 Electrostatics ...... 18 3.1 The Variational Method ...... 7 8.4 Magnetism ...... 18 3.2 Perturbation Theory ...... 7 9 Molecular&Spectroscopic Data 19 4 Electron Spin 7 10 Other Useful Data 20 5 Polylectronic Atoms 7 10.1 SI Units & Unit Prefixes ...... 20 5.1 Hamiltonian & Wavefunctions ...... 7 10.2 Energy Conversion Factors ...... 20 5.2 Angular Momenta ...... 8 10.3 Symbols for Elementary Particles ...... 20 5.3 Atomic Term Symbols & Spectra ...... 8 10.4 Fundamental Physical Constants ...... 21 10.5 Atomic Units ...... 21 6 Molecular Electronic Structure 9 10.6 The Greek Alphabet ...... 21 6.1 Molecular Orbital Theory ...... 9 10.7 Periodic Table of the Elements ...... 22 6.2 Molecular Term Symbols ...... 9 10.8 Atomic Masses for Selected Isotopes ...... 22 6.3 H¨uckel MO Theory ...... 9 10.9 Effective Nuclear Charge (Zeff = Z σ) .... 22 6.4 Molecular Symmetry ...... 10 − 6.5 The Born-Oppenheimer Approximation .... 10
1 1 Quantum Mechanics Orthonormalization of functions and Kronecker delta: 1.1 Formulae of Quantum Mechanics 0 for i = j ψ∗ψ dτ = δ , δ = (13) i j ij ij 1 for i =6 j Speed of electromagnetic waves in terms of frequency ν Z and wavelength λ: c = νλ (1) The sum and the difference of two operators Aˆ and Bˆ:
Frequency ν and wavelength λ with relation to wavenum- (Aˆ Bˆ)f(x)= Afˆ (x) Bfˆ (x) (14) bersν ˜: 1 ν ± ± ν˜ = = (2) λ c The product of two operators Aˆ and Bˆ: Planck quantization of energy: ˆ ˆ ˆ ˆ E = hν (3) ABf(x)= A[Bf(x)] (15) de Broglie wavelength λ in terms of the particle’s momen- Linear operators should satisfy: Identities of commuta- tum p: h h tors (where k is a constant and the operators are assumed λ = = p mv to be linear): Time-dependent Schr¨odinger equation for one-particle [A,ˆ Bˆ]= [B,ˆ Aˆ] (16) (with mass m), one-dimensional system, in terms of wave- − [A,ˆ Aˆn] = 0, n = 1, 2, 3,... (17) function Ψ and potential energy function V (x,t)(~ = h ): 2π ˆ ˆ ˆ ˆ ˆ ˆ ~ ∂Ψ(x,t) ~2 ∂2Ψ(x,t) [kA, B]=[A,kB]= k[A, B] (18) = + V (x,t)Ψ(x,t) (4) − i ∂t −2m ∂x2 [Aˆ + B,ˆ Cˆ]=[A,ˆ Cˆ]+[B,ˆ Cˆ] (19) The probability at time t of finding the particle in the [A,ˆ Bˆ + Cˆ]=[A,ˆ Bˆ]+[A,ˆ Cˆ] (20) region of the x-axis lying between x and x + dx, for a ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ one-particle, one-dimensional system: [AB, C]=[A, C]B + A[B, C] (21) Ψ(x,t) 2 dx (5) [A,ˆ BˆCˆ]=[A,ˆ Bˆ]Cˆ + Bˆ[A,ˆ Cˆ] (22) | | The function Ψ(x,t) 2 is the probability density, de- Average value of an observable A: fined as | | Ψ 2 =Ψ∗Ψ (6) | | ∗ ∗ A = ψ Aψˆ dτ (23) with Ψ being the complex conjugate of the function Ψ, h i the complex conjugate being formed by replacing i with Z i wherever it occurs. If A and B are any two properties and k is a constant: − A + B = A + B and kA = k A (24) Time-dependent wave function Ψ is a function of time- h i h i h i h i h i independent wavefunction ψ(x) and time function f(t): ~ Ψ(x,t)= ψ(x) f(t)= ψ(x) e−iEt/ (7) However, the average value of a product need not equal the product of the average values: Time-independent Schr¨odinger equation in terms of the ˆ AB = A B (25) Hamiltonian operator H, the wavefunction ψ (or h i 6 h ih i eigenfunction), and energy E (or eigenvalue): pˆ2 Hermitian operators are defined as: Hψˆ = Eψ, Hˆ = Tˆ + V,ˆ Tˆ = q (8) 2m ∗ ˆ ˆ ∗ ψmAψndτ = ψn(Aψm) dτ (26) Linear momentum quantum mech. operator in the q axis: Z Z ~ ∂ pˆq = (9) The variance σ2 and the standard deviation σ (or ∆) i ∂q of an observable A: One-particle three-dimensional Hamiltonian operator: 1/2 ~2 ∂2 ∂2 ∂2 2 2 2 2 2 ˆ 2 2 σA = A A and σA ∆A = A A H = +V (x,y,z), = 2 + 2 + 2 (10) h i−h i ≡ h i−h i −2m∇ ∇ ∂x ∂y ∂z The probability of finding a particle between x1 and x2: Robertson inequality x2 ∗ Probability (x1 x x2)= ψ (x)ψ(x) dx (11) 1 ∗ ˆ ˆ ≤ ≤ σAσB ∆A∆B Ψ [A, B]Ψ dτ Zx1 ≡ ≥ 2 Z
The normalization condition in three-dimension: +∞ +∞ +∞ Heisenberg uncertainty relations: position x and linear ∗ ψ (x,y,z)ψ(x,y,z) dxdy dz momentum px, energy E and time t, and angle φ and an- Z−∞ Z−∞ Z−∞ gular momentum L complementaries: = ψ∗ψdτ = 1 (12) ~ ~ ~ ∆x∆p , ∆E∆t , ∆φ∆L (27) Z x ≥ 2 ≥ 2 z ≥ 2
2 ∗ ˆ 1.2 Postulates of Quantum Mechanics The definite integral Ψ1AΨ2dτ is abreviated: Postulate 1 (Wavefunctions): The state of a quantum Ψ Aˆ Ψ 1 Aˆ 2 Ψ∗AˆΨ dτ A (31) h 1| | 2i≡h |R | i≡ 1 2 ≡ 12 mechanical system is completely specified by a state func- An operator Aˆ is said to be HermitianZ if it satisfies: tion, Ψ(r, t), that is a function of the coordinates of the ψ Aˆ ψ = ψ Aˆ ψ ∗ ψ∗ Aψˆ dτ = ψ (Aψˆ )∗dτ particles r and the time t. If time is not a variable, its h m| | ni h n| | mi ≡ m n n m Z Z state is completely specified by a time-independent wave- 1.4 Theorems of Quantum Mechanics function ψ(r). All possible information about the system Hermitian operators: can be derived from Ψ(r, t). These wavefunctions are well behaved: single-valued, continuous, and quadratically in- 1. The eigenvalues of a Hermitian operator are real. ˆ tegrable. 2. Two eigenfunctions of a Hermitian operator A that correspond to different eigenvalues are orthogonal. Postulate 2 (Operators): To every observable in classical Eigenfunctions of Aˆ that belong to a degenerate mechanics there corresponds a linear Hermitian quantum eigenvalue can always be chosen to be orthogonal. mechanical operator. The operator is obtained from the 3. Let the functions g1 , g2 , ... be the complete set of classical mechanical expression for the observable written eigenfunctions of the Hermitian operator Aˆ, and let in terms of Cartesian coordinates and corresponding lin- the function Ψ be an eigenfunction of Aˆ with eigen- ear momentum components by replacing each coordinate value k. Then if Ψ is expanded as Ψ = aigi q by itself and the corresponding momentum component i ≡ Ψ = i gi gi Ψ , the only nonzero coefficients ai pq by i~ ∂/∂q. | i | ih | i P − are those for which gi has the eigenvalue k. Postulate 3 (Eigenvalues): In any measurement of the These threeP theorems can be summarized as: the eigen- physical observable A associated with the operator Aˆ, the functions of a Hermitian operator form a complete, or- only values that will ever be observed are the eigenvalues thonormal set, and the eigenvalues are real. ai, which satisfy the eigenvalue equation Commuting operators: AˆΨi = aiΨi 4. If the linear operators Aˆ and Bˆ have a common com- ˆ ˆ The eigenfunctions Ψi are required to be well-behaved. plete set of eigenfunctions, then A and B commute. Postulate 4 (Completeness): If Aˆ is a linear Hermitian 5. If [A,ˆ Bˆ]=0(Aˆ and Bˆ Hermitian), we can select a operator that represents a physically observable property, common complete set of eigenfunctions for them. then the eigenfunctions gi of Aˆ form a complete set. 6. If gm and gn are eigenfunctions of the Hermitian operator Aˆ with different eigenvalues (that is, if Postulate 5 (Average values): If Ψ(r, t) is the normal- Agˆ = a g and Agˆ = a g with a = a ), ized state function of a system at time t, then the average m m m n n n m n and if the linear operator Bˆ commutes with A6ˆ, then value of the observable A corresponding to Aˆ is g Bˆ g = 0 for a = a . h n| | mi n 6 m a = Ψ∗AˆΨ dτ Parity: h i Z The parity operator Πˆ replaces each Cartesian coordinate Postulate 6 (Time dependence): The wavefunction of a with its negative: Πˆf(x,y,z)= f( x, y, z). The eigen- system changes with time according to the time-dependent functions of the parity operator −Πˆ are− all− possible well- Schr¨odinger equation behaved even and odd functions. When the potential en- ∂Ψ(r, t) Hˆ Ψ(r, t)= i~ ergy V is an even function, the parity operator commutes ∂t with the Hamiltonian: [H,ˆ Π]ˆ = 0. ˆ where H is the Hamiltonian operator for the system. 7. When the potential energy V is an even function, Postulate 7 (Spin): The wavefunction of a system of elec- we can choose the stationary-state wave functions trons (fermions) must be antisymmetric to the interchange so that each ψi is either an even or an odd function. of any two electrons (fermions). Measurement: 8. If am is a nondegenerate eigenvalue of the operator 1.3 Dirac Notation (Bracket Notation) Aˆ, where Agˆ m = amgm, then, when the property A is measured in a system whose state function is A physical state is represented by a state vector called a Ψ, the probability of getting the result a is given ket and is denoted by Ψ . An operator acts on a ket from m by c 2, where c is the coefficient of g in the the left Aˆ Ψ . To every| keti Ψ , there exist a bra, denoted m m m expansion| | Ψ = c g . If the eigenvalue a is de- by Ψ in the| i bra space dual| toi the ket space. The bra dual i i i m generate, the probability of obtaining a when A to ch Ψ| is postulated to be c∗ Ψ (c is a complex number). m is measured is foundP by adding the c 2 values for An |inneri product of a brah and| a ket, representing two i those eigenfunctions whose eigenvalue| is| a . square integrable functions Ψ and Ψ is defined as, m 1 2 However, the probability of finding the nondegenerate ∗ 2 2 Ψ1 Ψ2 1 2 Ψ1Ψ2dτ (28) eigenvalue a in a measurement of A is c = g Ψ . h | i≡h | i≡ m | m| |h m| i| Z The quantity g Ψ 2 is called a probability amplitude. If with Ψ Ψ = Ψ Ψ ∗, and if Ψ is a third function: |h m| i| h 1| 2i h 2| 1i | 3i the property B has a continuous range of eigenvalues, the ∗ summation in the expansion Ψ = c g is replaced by an Ψ1 cΨ2 = c Ψ1 Ψ2 , cΨ1 Ψ2 = c Ψ1 Ψ2 (29) i i i 2 h | i h | i h | i h | i integration over the values of b and gb Ψ is interpreted Ψ Ψ +Ψ = Ψ Ψ + Ψ Ψ (30) as a probability density. P|h | i| h 3| 1 2i h 3| 1i h 3| 2i
3 2 Systems with Exact Solutions 2.3 The Harmonic Oscillator 2.1 The Particle in a Box The Schr¨odinger equation for the quantum mechanical The Schr¨odinger equation for a particle of mass m in a 1D one-dimensional harmonic oscillator QMHO of mass m is box of length l with potential-energy V = for x < 0 d2ψ 2mE ∞ + α2x2 ψ = 0 (37) and x>l, and V = 0 for 0 x l: 2 ~2 ≤ ≤ dx − ~2 d2ψ(x) = Eψ(x) (32) where α is a constant defined as − 2m dx2 2πνm km 1/2 The energy values En are α = = (38) ~ ~2 h2n2 E = n = 1, 2, 3 ... (33) n 8ml2 where k, the force constant, is defined is terms of ω, an- The wavefunctions (orthonormal, no. of nodes = n 1): gular frequency, and ν, the frequency, in such a way that − 1/2 1/2 2 nπx for 0 x l ω 1 k ψn(x)= sin ≤ ≤ (34) ν = = (39) l l and n = 1, 2, 3 ... 2π 2π m The Schr¨odinger equation for a particle of mass m moving The allowed energy levels Ev are ~2 2 in a 3D box of lengths a, b, and c is: 2m ψ(x,y,z) = 1 − ∇ Ev =(v + 2 )hν v = 0, 1, 2,... (40) Eψ(x,y,z). The allowed energy levels Enxny nz are: The general normalized wavefunctions for the QMHO are 2 n = 1, 2, 3,... h2 n2 n n2 x E = x + y + z n = 1, 2, 3,... 1/2 −αx2/2 nxny nz 8m a2 b2 c2 y ψv(x)= NvHv(α x)e (41) ! nz = 1, 2, 3,... The normalized wavefunctions: where Nv is the normalization constant defined as 1/4 1/2 1 α 8 nxπx nyπy nzπz Nv = v 1/2 (42) ψnxny nz (x,y,z)= sin sin sin (2 v!) π abc a b c 1/2 The degeneracy g for a particle in a cube (a = b = c): and Hv(α x) are Hermite polynomials, defined by: 2 2 2 n −x2 nxnynz nx + ny + nz nx,ny,nz g 2 d e H (x)=( 1)nex (43) 331 19 (3,3,1)(3,1,3)(1,3,3) 3 n − dxn 411 18 (4,1,1)(1,4,1)(1,1,4) 3 and obey the recurrence relation: 322 17 (3,2,2)(2,3,2)(2,2,3) 3 1 321 14 (3,2,1)(3,1,2)(2,3,1) 6 xHn(x)= nHn−1(x)+ 2 Hn+1(x) (44) (1,3,2) (1,2,3) (2,1,3) The first few Hermite polynomials are: 222 12 (2,2,2) 1 311 11 (3,1,1)(1,3,1)(1,1,3) 3 H (x) = 1 H (x) = 8x3 12x 0 3 − 221 9 (2,2,1)(2,1,2)(1,2,2) 3 4 2 H1(x) = 2x H4(x) = 16x 48x + 12 211 6 (2,1,1)(1,2,1)(1,1,2) 3 − H (x) = 4x2 2 H (x) = 32x5 160x3 + 120x 111 3 (1,1,1) 1 2 − 5 − The QMHO wavefunctions for the lowest four levels are 2.2 The Particle in a Ring α 1/4 2 The Schr¨odinger equation for a particle in a ring (or 2D ψ = e−αx /2 0 π rigid rotor) of radius r and moment of inertia I: 1/4 2 3 d Φ 4α −αx2/2 + m2Φ(φ)=0 (35) ψ1 = xe 2 π dφ 1/4 m is a quantum number (defined below). The energies: α 2 −αx2/2 m2~2 ψ2 = (2αx 1)e E = m = 0, 1, 2, 3,... (36) 4π − 2I ± ± ± 3 1/4 α 3 −αx2/2 The normalized wavefunctions are: ψ3 = (2αx 3x)e 9π − 1 imφ The number of nodes in the ψ’s QMHO = ν. The QMHO Φm(φ)= 1/2 e m = 0, 1, 2, 3,... (2π) ± ± ± ψs are orthogonal, and are either even or odd functions: a a m Complex form Real form f(x) dx = 2 f(x) dx f(x) even [f(x)= f( x)] 1 1 −a 0 − 1 Φ0 = Φ0 = Z Z √2π √2π a 1 iφ 1 cos φ f(x) dx = 0 f(x) odd [f(x)= f( x)] +1 Φ1 = e Φx = (Φ1 +Φ−1) = − − √2π √2 √π Z−a 1 −iφ 1 sin φ 1 Φ−1 = e Φy = (Φ1 Φ−1) = The vibrations of diatomic molecules can be approximated − √2π √2 − √π using QMHO, where m is replaced by the reduced mass 1 i2φ 1 cos2φ +2 Φ2 = e Φx2−y2 = (Φ2 +Φ−2) = µ = m1m2/(m1 + m2) in all relevant formulas. The al- √2π √2 √π lowed vibrational absorptions (selection rule): ∆ν = 1, 1 −i2φ 1 sin2φ 2 Φ−2 = e Φxy = (Φ2 Φ−2) = − √2π √2 − √π with the condition of having a nonzero dipole moment.
4 2.4 Angular Momentum 2.5 The Rigid Rotor The angular momentum operators in Cartesian and spher- The Hamiltonian operator of a rigid rotor with a moment ical coordinates are: of inertia I = µr2 is: ˆ 1 ˆ2 ∂ ∂ ∂ ∂ H = L (50) Lˆ = i~ y z = i~ sin φ + cot θ cos φ 2I x − ∂z − ∂y ∂θ ∂φ The rigid rotor eigenfunctions are the spherical harmonics Y m(θ,φ), where J rather than l is used for the rotational ∂ ∂ ∂ ∂ J Lˆ = i~ z x = i~ cos φ cot θ sin φ angular-momentum quantum numberr, and the egenval- y − ∂x − ∂z − ∂θ − ∂φ ues are ~2J(J + 1) ∂ ∂ ∂ E = J = 0, 1, 2,... (51) Lˆ = i~ x y = i~ J 2I z − ∂y − ∂x − ∂φ with degeneracy g = 2J + 1. The allowed pure-rotational Lˆ2 = Lˆ 2 = Lˆ Lˆ = Lˆ2 + Lˆ2 + Lˆ2 transitions for diatomic molecules (selection rule): ∆J = | | · x y z ∂2 ∂ 1 ∂2 1, with the condition of having a nonzero dipole moment. = ~2 + cot θ + The± frequencies of the pure-rotational spectral lines of a − ∂θ2 ∂θ sin2 θ ∂φ2 diatomic molecule are ν = 2(J + 1)B, where B = h/8π2I Commutation relations of angular momentum: and is called the rotational constant of the molecule. 2 [Lˆx, Lˆy]= i~Lˆz [Lˆ , Lˆx]=0 (45) 2 2.6 The Hydrogen (Hydrogenlike) Atom [Lˆy, Lˆz]= i~Lˆx [Lˆ , Lˆy]=0 (46) 2 Sch¨odinger Equation [Lˆz, Lˆx]= i~Lˆy [Lˆ , Lˆz]=0 (47) The Sch¨odinger equation for the hydrogenlike atom The spherical harmonics wavefunctions Y m(θ,φ) are l (atomic number Z) with an electron of mass m moves ˆ2 ˆ e eigenfunctions of both L and Lz operators: about a stationary nucleus of infinite mass, and with a Lˆ2Y m(θ,φ)= ~2l(l + 1)Y m(θ,φ) l = 0, 1, 2,... (48) Ze2 l l potential energy V (r)= 4πε r , is − 0 2 ˆ m ~ m 2 ∂ 2 ∂ψ 2 2 Ze LzYl (θ,φ)= mYl (θ,φ) m = 0, 1, 2,... (49) ~ r +Lˆ ψ 2µr + E ψ = 0 (52) ± ± − ∂r ∂r − 4πε r ˆ2 2 ~2 ˆ 0 The eigenvalues of L are L = l(l +1), and that of Lz where are L = ~m. The Y m(θ,φ) functions are defined as: z l m m 1 imφ ψnlm(r,θ,φ)= Rnl(r)Yl (θ,φ) (53) Y (θ,φ)= Sl,m(θ)Φm(φ)= Sl,m(θ)e l √2π where R(r) is the hydrogenlike radial wavefunction and 1/2 m 2l + 1 (l m )! Yl (θ,φ) are the spherical harmonics (the angular part). |m| 2 2 Sl,m(θ)= −| | P (cos θ) The radial Sch¨odinger equation is (a = 4πε ~ /m e ): 2 (l + m )! l 0 0 e |m| | | associated Legendre functions 2 8πε0E 2Z l(l + 1) where Pl (cos θ) are the . R′′ + R′ + + R = 0 The first few associated Legendre functions: r ae2 ar − r2 0 0 1 2 For H atom, and all central-force systems [V = V (r)]: P0 = 1 P2 = 2 (3cos θ 1) − ˆ ˆ2 ˆ ˆ ˆ2 ˆ P 0 = cos θ P 1 = 3sin θ cos θ [H, L ]=[H, Lz]=[L , Lz] = 0 1 2 − P 1 = sin θ P 2 = 3sin2 θ Quantum Numbers 1 − 2 1. Principal quantum number (n): n = 1, 2, 3,... The first spherical harmonics Y m(θ,φ) are: l 2. Angular momentum quantum number (l): l = 0(s), 1(p), 2(d), 3(f), 4(g), 5(h),...,n 1. l m Spherical harmonic − 3. Magnetic quantum number (ml): 0 1 0 0 Y0 = m = 0, 1, 2,..., l. (4π)1/2 ± ± ± 1 1 1/2 4. Spin quantum number (ms): ms =+ 2 or 2 . 0 3 2 − 1 0 Y1 = cos θ The degeneracies g = 2n when spin vales are included. 4π 3 1/2 1 Y 1 = sin θ eiφ Hydrogenlike Wavefunctions 1 8π 1/2 The radial wavefunctions R (r) depend on two quantum 3 nl Y −1 θ e−iφ numbers n and l and are given by 1 1 = sin 1/2 − 8π 2Z 3 (n l 1)! 2Zr 1/2 l 2l+1 −Zr/na0 Rnl(r)= − − 3 r Ln+l e 0 5 2 − na0 2n[(n + l)!] na0 2 0 Y2 = (3 cos θ 1) ( ) 16π − 1/2 where the combinitorial factor in front is a normalization 1 15 iφ 2l+1 2Zr 1 Y2 = sin θ cos θ e constant; and L is known as an associated La- 8π n+l na0 1/2 guerre polynomial, the first few of them are: −1 15 −iφ 1 Y2 = sin θ cos θ e − 8π n = 1 l = 0 L1(x)= 1 1/2 1 − 2 15 2 2iφ n = 2 l = 0 L1(x)= 2!(2 x) 2 Y2 = sin θ e 2 32π 3 − − 1/2 l = 1 L3(x)= 3! −2 15 2 −2iφ 1 − 1 2 2 Y2 = sin θ e n = 3 l = 0 L3(x)= 3!(3 3x + 2 x ) − 32π 3 − − l = 1 L4(x)= 4!(4 x) l = 2 L5(x)= −5! − Spherical harmonics form an orthonormal set: 5 − 2π π ′ m ∗ m [Y (θ,φ)] Y ′ (θ,φ) sin θdθdφ = δ ′ δ ′ l l ll mm 5 Z0 Z0 The first few hydrogen radial wavefunctions Rnl(r) are: The real complete normalized hydrogenlike atomic wave- 3/2 functions for n = 1, 2, and 3 are: Z −Zr/a 1s R10(r) = 2 e 0 3/2 a0 1 Z −Zr/a0 3/2 ψ1s = e 1 Z Zr −Zr/2a 2s R20(r)= 1 e 0 √π a0 √ a − 2a 2 0 0 3/2 5/2 1 Z Zr −Zr/2a 1 Z −Zr/2a 0 2p R21(r)= re 0 ψ2s = 2 e √ a 4√2π a0 − a0 2 6 0 2 Z 3/2 2Zr 2Z2r2 5/2 −Zr/3a0 1 Z −Zr/2a 3s R30(r)= 1 + 2 e ψ = re 0 cos θ 3√3 a0 − 3a0 27a 2pz 0 4√2π a0 8 Z 3/2 Zr Z2r2 −Zr/3a0 5/2 3p R31(r)= 2 e 1 Z 27√6 a0 a0 − 6a −Zr/2a0 0 ψ2px = re sin θ cos φ 7/2 4√2π a0 4 Z 2 −Zr/3a 3d R32(r)= r e 0 5/2 81√30 a0 1 Z −Zr/2a0 ψ2py = re sin θ sin φ 4√2π a0 The probability of finding the electron between r and 1 Z 3/2 Zr Z2r2 r + dr: 2 2 −Zr/3a0 [Rnl(r)] r dr (54) ψ3s = 27 18 + 2 2 e 81√3π a0 − 2a0 a The function [R(r)]2 r2, the radial distribution func- 0 21/2 Z 5/2 Zr tion, determines the probability of finding the electron at ψ = 6 re−Zr/3a0 cos θ 3pz 81√π a − a a distance r from the nucleus. 0 0 21/2 Z 5/2 Zr −Zr/3a0 The average distance for an electron may be expressed as: ψ3px = 6 re sin θ cos φ ∞ 81√π a0 − a0 ∗ 2 5/2 r = Rnl(r)rRnl(r) r dr (55) 21/2 Z Zr h i 0 ψ = 6 re−Zr/3a0 sin θ sin φ Z 3py The radial functions are shown in Fig. 1. 81√π a0 − a0 7/2 2 2 1 Z R(r) r [ R(r)] 2 −Zr/3a0 2 ψ3dz2 = r e (3cos θ 1) 81√6π a0 − 21/2 Z 7/2 2 −Zr/3a0 1s 1s ψ3dxz = r e sin θ cos θ cos φ 81√π a0 0 5 10 15 20a 1a 0 0 21/2 Z 7/2 2 −Zr/3a0 ψ3dyz = r e sin θ cos θ sin φ 81√π a0 2s 2s 1 Z 7/2 2 −Zr/3a0 2 ψ3dx2−y2 = r e sin θ cos 2φ 81√2π a0 1 Z 7/2 2 −Zr/3a0 2 2p 2p ψ3dxy = r e sin θ sin2φ 81√2π a0 The ψnlm functions are called one-electron functions. The functions ψnlm(r,θ,φ) are all mutually orthonormal: 3s 3s ∞ 2π π ∗ ′ ′ ′ 2 ψnlm(r,θ,φ)ψn l m (r,θ,φ)r sin θdrdθdφ Z0 Z0 Z0 = δnn′ δll′ δmm′ (56) 3p 3p The number of radial nodes in Rnl(r) = n l 1. The number of angular nodes = l. Thus the total− number− of nodes in a hydrogenlike wavefunction ψnlm(r,θ,φ)= n 1. 3d 3d −
2 2 Energy Levels Figure 1: Radial functions Rnl(r) and r [Rnl(r)] .
The complete normalized hydrogenlike atomic wavefunc- Bohr relation for energy levels (En) of a Hydrogenlike m tions ψnlm(r,θ,φ) = Rnl(r)Yl (θ,φ) for n = 1 and 2 are: atom with charge Z and bohr radius a0: 2 4 2 2 2 3/2 meZ e Z e ~ 4πε0 1 Z −Zr/a E = = , a = ψ ψ e 0 n 2 2 2 2 0 2 100 = 1s = −2(4πε0) ~ n −8πε0a0n mee √π a0 Z 3/2 Zr Rydberg energy of an electronic level in wavenumbers is 1 −Zr/2a0 ψ200 = ψ2s = 1 e 2 2 √π 2a0 − 2a0 ν˜ = RHZ /n . Rydberg formula for the emission radia- − Z 5/2 tion in wavenumbers of a Hydrogenlike atom with charge 1 −Zr/2a0 ψ210 = ψ2p0 = re cos θ √π 2a0 Z (n2 > n1): 5/2 1 ν E E 1 1 1 Z 2 1 2 −Zr/2a0 iφ ν˜ = = = − = R∞Z 2 2 ψ211 = ψ2p1 = re sin θe λ c hc n − n 8√π a0 1 2 Z 5/2 5 −1 1 −Zr/2a0 −iφ where R∞ = 1.097373157 10 cm is the Rydberg con- ψ21−1 = ψ2p−1 = re sin θe × 5 −1 8√π a0 stant ( R = 1.096776 10 cm ). ≈ H ×
6 3 Approximation Methods 4 Electron Spin Commutation relations of spin angular momentum: 3.1 The Variational Method 2 [Sˆx, Sˆy]= i~Sˆz [Sˆ , Sˆx] = 0 The variational theorem for a trial wavefunction (φ): 2 [Sˆy, Sˆz]= i~Sˆx [Sˆ , Sˆy] = 0 φ∗Hφdτˆ 2 Egs Eφ = (57) [Sˆ , Sˆ ]= i~Sˆ [Sˆ , Sˆ ] = 0 ≤ φ∗φdτ z x y z R The magnitude of the spin angular momentum vector S φ may be constructed by a linear combination of n linearly R n is: √ 2 ~ 1 independent basis functions: φ = i=1 cifi, where ci S = S = S = s(s + 1) , s = 2 (66) are variational parameters. The values of c that give where|s|is the spin quantum number. S , the component i p z the minimum energy satisfy the followingP set of simulta- of the spin angular momentum in z direction, can take on neous linear algebraic equations: only the values 1 Sz = ms~, ms = (67) c1(H11 ES11)+c2(H12 ES12)+ +cn(H1n ES1n)=0 ± 2 − − · · · − where ms is the quantum number for the z component of c1(H21 ES21)+c2(H22 ES22)+ +cn(H2n ES2n)=0 (58) − − · · · − the spin. See the table bellow for an analogy between or- ··················································· ····· bital and spin angular momenta for a single electron. c1(Hn1 ESn1)+c2(Hn2 ESn2)+ +cn(Hnn ESnn)=0 − − · · · − Orbital Angular Spin Angular Momentum Momentum where Hij and Sij are called matrix elements defined as Angular momentum vector L S ∗ ˆ ∗ Hij = φi Hφi dτ and Sij = φi φi dτ (59) Magnitude of above L = l(l + 1)~ S = s(s + 1)~ z Component of angular L = m ~ S = m ~ Z Z ˆ zp l zp s where Sij is called the overlap integral. If H is Hermitian momentum vector ˆ ∗ ˆ2 ˆ2 and fi, fj, and H are real, then Hij = Hji = Hji and Operator for square of L S ∗ angular momentum Sij = Sji = Sji. The system of equations (58) are solved Operator for z component Lˆ Sˆ for E using the secular determinant, det(Hij ESij )=0: z z − of angular momentum H11 ES11 H12 ES12 H1n ES1n − − · · · − Quantum number l = 0, 1, 2,... s = 1 H21 ES21 H22 ES22 H2n ES2n 2 − − · · · − Quant. no. for z comp. m = 0, 1, 2,..., l m = 1 . . . . = 0 (60) l s 2 . . .. . ± ± ± ± . . . spin eigenfunctions The , α and β, are orthonormal: Hn1 ESn1 Hn2 ESn2 Hnn ESnn − − · · · − ∗ ∗ ∗ ∗ The secular equation associated with this secular determi- α αdσ = β β dσ = 1, α β dσ = β αdσ = 0 ˆ2 nat is an nth degree polynomial in E. With E determined whereZ σ is the spin variable.Z The operators S and Sˆz ˆ2 as a root of the secular equation (60), the ci can be found commute with Hˆ , L , and Lˆz. Moreover: by solving n 1 of the set of equations (58) (see 7.11). Sˆ2α = 1 ( 1 + 1)~2α = 3 ~2α, Sˆ2β = 1 ( 1 + 1)~2β = 3 ~2β − § 2 2 4 2 2 4 Sˆ α =+ 1 ~α, Sˆ β = 1 ~β 3.2 Perturbation Theory z 2 z − 2 In a perturbation approximation, one divides Hˆ into two 5 Polylectronic Atoms parts: ˆ ˆ 0 ˆ ′ H = H + λH (61) 5.1 Hamiltonian & Wavefunctions where Hˆ 0, the unperturbed system, can be solved exactly, The Hamiltonian for an atom containing n electrons: Hˆ ′ is the perturbation, and the system with Hamiltonian n n n−1 n Hˆ = Hˆ 0 + λHˆ ′ is the perturbed system. When λ is zero, ~2 Ze2 e2 Hˆ = 2 + (68) we have the unperturbed system. As λ increases, the per- −2m ∇i − 4πε r 4πε r e i=1 i=1 0 i i=1 j=i+1 0 ij turbation grows larger, and at λ = 1 the perturbation is where an infinitelyX heavyX point nucleusX wasX assumed. For full. The wave function ψn and energy En of state n of a neutral atom, Z = n. The restriction j = i + 1 avoids the perturbed system can be written as counting each interelectronic repulsion twice and avoids (0) (1) 2 (2) k (k) 2 ψn = ψn + λψn + λ ψn + + λ ψn + (62) e /4πε0rii terms. The Hamiltonian (68) is incomplete, ··· ··· because it omits spin-orbit and other interactions. E = E(0) + λE(1) + λ2E(2) + + λkE(k) + (63) n n n n ··· n ··· The Rayleigh-Schrodinger first-order energy correction is A wavefunction for the ground state of a polyelectronic (1) (0)∗ ˆ ′ (0) (0) ˆ ′ (0) ˆ ′ atom with n electrons is given by a Slater determinant: En = ψn H ψn dτ = ψn H ψn = Hnn (64) h | | i φ1(1)α(1) φ1(1)β(1) φ2(1)α(1) φm(1)β(1) Z (0) (1) (0) ′ ··· and E E + E = E + Hˆ . The first-order φ1(2)α(2) φ1(2)β(2) φ2(2)α(2) φm(2)β(2) n n n n nn 1 ··· wavefunction≈ correction is . . . . . (69) √n! ...... (0)∗ ˆ ′ (0) (0) ˆ ′ (0) (1) ψm H ψn dτ (0) ψm H ψn (0) ψ = ψ = ψ φ1(n)α(n) φ1(n)β(n) φ2(n)α(n) φm(n)β(n) n (0) (0) m h (0)| | (0) i m ··· m6=n R En Em m6=n En Em where m = n/2 if n is even and m =(n + 1)/2 if n is odd. X − X − The second-order energy correction is The factor 1/√n! is a normalization factor. φ ,φ , ,φ 1 2 ··· m (0)∗ ˆ ′ (0) 2 (0) ˆ ′ (0) 2 are spatial hydrogenlike orbitals, i.e, φ1 = 1s, φ2 = 2s, (2) ψm H ψn dτ ψm H ψn φ3 = 2px etc. The functions φiα and φiβ are called spin En = | (0) (0) | = |h (0)| | (0)i| m6=n R En Em m6=n En Em orbitals. All the elements in a given column of a Slater X − X − determinant involve the same orbital, whereas elements in ′ 2 (0) ′ Hmn the same row all involve the same electron. A shorthand En En + Hnn + (0)| | (0) (65) ≈ En Em notation for (69) is: ψ = 1s1s2s . mX6=n − | ···| 7 OrbitalAngularMomentum SpinAngularMomentum TotalAngular Electron Atom Electron Atom MomentumofAtom
Angular li L = li si S = si J = L + S momentum vector P P
z comp. of lzi Lz = lzi szi Sz = szi Jz = Lz + Sz ang. mom. vector P P 1 Quantum li(0, 1, 2,...) L = l1 + l2,l1 + l2 1, si( ) S = s1 + s2,s1 + s2 1, J = L + S,L + S 1, − 2 − − number ..., l1 l2 ..., s1 s2 ..., L S | − | | − | | − | 1 Quantum number mli( li,..., ML = mli msi( ) MS = msi MJ = ML + MS − ± 2 P P for z comp. +li) ( L,..., +L) ( S,..., +S) ( J,..., +J) − − −
5.2 Angular Momenta A set of equal-energy states with the same L and S con- stitutes a term, which is denoted by the term symbol The set of quantum numbers, L, ML, S, MS are desig- nated for multielectron atoms. Multielectron atoms have 2S+1L total orbital angular momentum L and total spin angular where 2S +1 is spin multiplicity and L is a code letter: 2 momentum S (See the Table above). Operators Lˆ , Lˆz, 2 L value 0 1 2 3 4 5 6 7 8 9 10 Sˆ , and Sˆz commute with Hˆ . These operators yield: 2 2 Code Letter S P D F G H I K L M N Lˆ ψ = ~ L(L + 1)ψ, Lˆzψ = ~MLψ The degeneracy of a term is (2L + 1)(2S + 1). When Sˆ2ψ = ~2S(S + 1)ψ, Sˆ ψ = ~M ψ z S spinorbit interaction is included, each term is split into a Addition of angular momenta: Let M1 and M2 be number of levels, each having a different value of J, where two angular momenta with quantum numbers j1, m1 and J ranges from L + S to L S. The symbol for a level is j , m , and let M be their sum: M = M + M . For the − 2 2 1 2 2S+1L angular-momentum sum, Mˆ 2 has eigenvalues J(J + 1)~2 J ˆ 2 ~ Each level is (2J + 1)-fold degenerate, corresponding to and Mz has eigenvalues MJ , where the possible values the (2J + 1) values of M , which range from J to J. of J and MJ are: J − J = j1 + j2,j1 + j2 1,..., j1 j2 (70) − | − | In summary, if we consider the Hamiltonian (68) without M = J, J 1,..., J (71) J − − the interelectronic repulsion term, all atomic states corre- 5.3 Atomic Term Symbols & Spectra sponding to the same electronic configuration are degener- The possible L and S values arising from some configura- ate. Adding the interelectronic term to the Hamiltonian, tions are found by consulting the Table below. we lift the degeneracy between states with different L or S or both, thus splitting each configuration into terms. Configuration Terms Next, we add in spin-orbit interaction, which splits each Equivalent el term into levels. Each level is composed of states with the s2, p6, d10 1S s1 2S same value of J. The degeneracy of each level is removed p1, p5 2P by applying an external field. To conclude schematically: 2 4 3 1 1 p , p P, D, S interelect spin-orbit external 3 4 2 2 Configurations Terms Levels States p S, D, P −−−−−−→repulsions −−−−−−−→interaction −−−−−→field d1, d9 2D 2 8 3 3 1 1 1 The empirical Hund’s rules are: d , d F, P, G, D, S d3, d7 4F, 4P, 2H, 2G, 2F, 2D, 2D, 2P 1. The term arising from the ground configuration with d4, d6 5D, 3H, 3G, 3F, 3F, 3D, 3P, 3P, 1I, the maximum multiplicity (2S +1) lies lowest in E. 1 1 1 1 1 1 1 G, G, F, D, D, S, S 2. For levels with the same multiplicity, the one with d5 6S, 4G, 4F, 4D, 4P, 2I, 2H, 2G, 2G, the maximum value of L lies lowest in E. 2F, 2F, 2D, 2D, 2D, 2P, 2S Nonequiv el 3. For levels with the same S and L, two cases: 1 1 3 1 1 s s S, S a) subshell <2 filled, state with lowest J most stable 1 1 3 1 1 s p P, P b) subshell>2 filled, state with highest J most stable s1 d1 3D, 1D Atomic Spectra and Selection Rules 1 1 3 1 3 1 3 1 p p D, D, P, P, S, S The selection rules for hydrogenlike atoms are: (1) ∆n, p1 d1 3F, 1F, 3D, 1D, 3P, 1P 1 1 3 1 3 1 3 1 3 1 3 1 unrestricted, (2) ∆l = 1, (3) ∆ml = 0, 1. The selec- d d G, G, F, F, D, D, P, P, S, S tion rules for many-electron± light atoms are:± s1 s1 s1 4S, 2S, 2S 1 1 1 4 2 2 1. ∆L = 0, 1, (except 0 0 not allowed). s s p P, P, P ± → s1 p1 p1 4D, 2D, 2D, 4P, 2P, 2P, 2P, 4S, 2S, 2S 2. ∆l = 1, when 1 e is promoted from the gs config. ± s1 p1 d1 4F, 2F, 2F, 4D, 2D, 2D, 4P, 2P, 2P 3. ∆J = 0, 1, (except 0 0 not allowed). ± → 4. ∆S = 0. 8 6 Molecular Electronic Structure 6.3 H¨uckel MO Theory 6.1 Molecular Orbital Theory H¨uckel theory is a simple LCAO-MO theory of π electrons in conjugated and aromatic molecules whose σ skeleton is There are two theoretical models for the description of assumed planar. In H¨uckel MO theory, the secular equa- the electronic structure of molecules: Molecular Or- tion is simplified by making the following assumptions: bital (MO) theory and Valence Bond (VB) theory. In Sij = δij Hii = α Hij = βδi,i±1 (73) MO theory, molecular orbitals φ are expressed as a linear The results for a few simple molecules are tabulated below: combination of atomic orbitals (LCAO): φ = i ciχi. No. of Frontier H-L Molecule Energy nodes orbital energy gap 6.2 Molecular Term Symbols P Ethylene E2 = α β 1 LUMO 2 β − | | Diatomic Molecules: Diatomic MOs are classified ac- C2H4 E1 = α + β 0 HOMO Allyl cation E3 = α √2β 2 LUMO √2 β cording to λ. The letters σ,π,δ,φ,γ,... denote λ m − | | ≡ | | C H+ E = α 1 HOMO values of 0, 1, 2, 3, 4, ..., respectively, where m~ is the 3 5 2 E = α + √2β 0 Lˆ eigenvalue. Moreover, g or u denotes even or odd func- 1 z E4 = α 1.618β 3 − tions, and the star denotes an antibonding MO. Butadiene E3 = α 0.618β 2 LUMO 1.236 β − | | C4H6 E2 = α + 0.618β 1 HOMO 2S+1 Each diatomic-molecule term has the form: Λ, where E1 = α + 1.618β 0 E4 = α 2β 2 Λ is a code letter (Σ, Π, ∆, Φ, Γ,...) that gives the ML − value (0, 1, 2, 3, 4,...). Σ terms are designated + or| |, Cyclobuta- E3 = α 1 SOMO 0 diene C H E = α 1 SOMO according to whether the eigenvalue of ψ for reflection− in 4 4 2 e E1 = α + 2β 0 a plane containing the molecular axis is +1 or 1. For E5 = α 1.618β 2 − − Cyclopenta- E4 = α 1.618β 2 LUMO 2.236 β homonuclear diatomics, a g or u subscript is added to the − | | dienyl anion E3 = α + 0.618β 1 HOMO term symbol to show whether ψe is even or odd. The Table C H− E = α + 0.618β 1 below lists terms arising from various configurations. 5 5 2 E1 = α + 2β 0 E6 = α 2β 3 Configuration Terms − 1 + 3 + E5 = α β 2 σσ, Σ , Σ − Benzene E4 = α β 2 LUMO 2 β σπ, σπ3 1Π, 3Π − | | 3 1 + 3 + 1 − 3 − 1 3 C6H6 E3 = α + β 1 HOMO ππ, ππ Σ , Σ , Σ , Σ , ∆, ∆ E2 = α + β 1 3 3 1 3 1 3 πδ, π δ, πδ Π, Π, Φ, Φ E1 = α + 2β 0 2 + σ Σ E6 = α 1.802β 5 − 2 4 4 1 + E5 = α 1.247β 4 σ , π , δ Σ − π, π3 2Π Hexatriene E4 = α 0.445β 3 LUMO 0.890 β C H E = α +− 0.445β 2 HOMO | | π2 1Σ+, 3Σ−, 1∆ 6 8 3 E2 = α + 1.247β 1 δ, δ3 2∆ E1 = α + 1.802β 0 δ2 1Σ+, 3Σ−, 1Γ HOMO/LUMO/SOMO: Highest occupied/lowest unoccupied/singly- occupied MOs. H-L: HOMO-LUMO. Polyatomic Molecules: As for diatomic molecules, the For a cyclic polyene of formula CN HN containing N car- electronic terms of polyatomic molecules are classified as bon atoms in the ring, the general solution of the secular singlets, doublets, triplets, ..., according to the value of determinant yields the following energy of the k-th level: 2S +1, provided spin-orbit interaction is omitted. For lin- 2kπ E = α + 2β cos (74) ear polyatomic molecules, the same term classifications are k N used as for diatomic molecules, giving such possibilities as where k = 0, 1, 2,..., (N 1)/2, N/2 for even N and 1Σ+, 1Σ−, 3Σ+, 1Π etc. For linear polyatomic molecules k = 0, 1, ±2,...,± (N ± 1)/−2 for odd N. For linear with a center of symmetry, the g, u classification is added. polyene± systems:± ± − (k + 1)π E = α + 2β cos (75) The electronic states of nonlinear polyatomic molecules k N + 1 where k = 0, 1,...,N 1. Frost (or polygon) diagrams for are classified according to the behavior of the electronic − wave function on application of the symmetry operators drawing out the energy levels in cyclic planar systems: A circle centered at α with radius 2β is inscribed with a poly- OˆR of the molecule (see 6.4). Each molecular electronic term of a molecules is designated§ by giving the irreducible gon with one vertex pointing down; the vertices represent representation of the electronic wave functions of the term, energy levels. See Fig 2. with the spin multiplicity 2S + 1 as a left superscript. 2β The letters A and B designate symmetry species of or- α bitally nondegenerate electronic terms. The following let- ter labels are used for the irreducible representation, ac- 2β cording to the orbital degeneracy n: 2β n 1 2 3 4 5 (72) Letter A, B E T G H α α 2β For most molecules in their electronic ground states the 2β electronic wave function is singlet and belongs to the (non- 1 degenerate) totally symmetric species (e. g. A1 for H2O Figure 2: Frost diagrams: pattern of energy levels in cyclic 1 and A1g for benzene). polyenes. Dashed lines are the nonbonding energy levels. 9 For compounds with π-system containing heteroatoms X, For the symmetry operation Rˆ that brings a point at x,y,z ′ ′ ′ modified parameters αX and βCX are used, defined as to x ,y ,z , the corresponding operator OˆR is defined as: ′ ′ ′ OˆRf(x ,y ,z ) = f(x,y,z). If a molecule has the symme- try operations Rˆ1, Rˆ2, ..., then the operators OˆR , OˆR , αX = α + hXβ (76) 1 2 ..., commute with the molecular Hamiltonian Hˆ . If Rˆ1, βCX = kCXβ (77) Rˆ2, ...all commute with one another, then the molecular ˆ ˆ wavefunctions are eigenfunctions of OR1 , OR2 , .... The hX and kCX values are listed in the Table below Symmetry Point Groups Element h k Boron B 1 C–B 0.7 The set of all symmetry operations of a molecule consti- − N–B 0.8 tutes a mathematical point group. For a crystal of infi- Carbon C 0 CC 1.0 nite extent, we can have symmetry operations that leave C–C 0.9 no point fixed, giving rise to space groups. A group is a C=C 1.1 set of elements (or members) and a rule which combines Nitrogen N· 0.5 C–N 0.8 any two elements to form a third element. N·· 1.5 CN 1 N+ 2 N–O 0.7 · The elements are said to form a group under the rule of Oxygen O 1 C–O 0.8 combination if the following four conditions are satisfied: O·· 2 C=O 1 (a) closure, (b) associativity, (c) the existence of an O+ 2.5 identity inverse Fluorine F 3 C–F 0.7 element, (d) the existence of an for Chlorne Cl 2 C–Cl 0.4 each element. Bromine Br 1.5 C–Br 0.3 CC and CN bonds are aromatic. The number of elements in a group is called the order of the group. The symmetry operations Aˆ, Bˆ, Cˆ, ...of a molecule form a group with the rule of combination for Bˆ 6.4 Molecular Symmetry and Cˆ being the product of the symmetry operations Bˆ and Cˆ. The product BˆCˆ means we first apply the opera- Symmetry Elements and Operations tion Cˆ to the molecule and we then apply Bˆ to the result found by applying Cˆ. A group for which BˆCˆ = CˆBˆ for A symmetry operation transforms an object into a po- every pair of group elements is commutative (also called sition that is physically indistinguishable from the original Abelian). position and preserves the distances between all pairs of points in the object. A symmetry element is a geomet- rical entity (point, line, or plane) with respect to which a 6.5 The Born-Oppenheimer Approxima- symmetry operation is performed. For molecules, the four tion kinds of symmetry elements are: The non-relativistic Hamiltonian operator Hˆ for a system
1. n-fold axis of symmetry (Cn): rotation about of M nuclei and n electrons described by position vectors an axis by (360/n)◦, n (integer) is the order of the RA and ri respectively, in atomic units is:
axis. The rotational axis of highest order, called the n M n M n n M M ˆ 1 2 1 2 ZA 1 ZAZB principal axis of rotation H = −X ∇i − X ∇A − X X + X X + X X , is often made the z axis. 2 2M r r R i=1 A=1 A i=1 A=1 iA i=1 j>i ij A=1 B>A AB Plane of symmetry | {zˆ } | ˆ{z } | ˆ{z } | {z } | ˆ{z } 2. (σ), reflection of all the nuclei Te TN VNe Vˆee VNN through a plane, A plane of symmetry containing the (78) principal axis of rotation is designated σv (for verti- where MA is the ratio of the mass of nucleus A to the cal); a plane of symmetry perpendicular to this axis mass of an electron. According to the Born-Oppenheimer is designated σh (for horizontal). approximation, (the electrons in a molecule are considered to be moving in the field of fixed nuclei), Tˆ = 0 and Vˆ 3. Center of symmetry (i), inversion of all the nuclei N NN term is a constant. The remaining terms in (78) are called through the center. the electronic Hamiltonian, 4. n-fold rotation–reflection axis of symmetry n n M n n ˆ 1 2 ZA 1 (Sn) (also called an improper axis), rotation by He = i + (79) − 2∇ − riA rij (360/n)◦ (n integer) about an axis, followed by re- i=1 i=1 A=1 i=1 j>i X X X X X flection in a plane perpendicular to the axis. Corresponding to Hˆe is the electronic wave function, ψe, ˆ The product of symmetry operations means successive and the electronic energy, Ee (Heψe = Eeψe). For fixed ˆn ˆ ˆ nuclei, the total energy E = E + V , where V is performance of them. We have Cn = E, where E is the tot e NN NN n n identity operation;σ ˆ = Eˆ and ˆi = Eˆ for even n, and M M n ˆn ˆ ˆ ˆ ˆ ZAZB σˆ =σ ˆ and i = i for odd n; also, S1 =σ ˆ, and S2 = i. VNN = (80) Two symmetry operations may or may not commute. RAB AX=1 B>AX
10 7 Mathematics Equations for conic sections and the sphere: 2 2 2 7.1 Algebra, Geometry & Trigonometry circle x + y = r radius r centered at the origin sphere x2 + y2 + z2 = r2 radius r centered at the origin Arithmetic operations and zero 2 y2 ellipse x + = 1 a: length of the semimajor axis a c ac a2 b2 b: length of the semiminor axis a(b + c)= ab + ac, = b · d bd parabola y = ax2 + b the vertex is at y = b a c ad bc a/b a d = ± , = hyperbola xy = constant b ± d bd c/d b · c if a = 0, then Trigonometric functions and properties a 6 0 = , = 0, a0 = 1, 0a = 0, 0! = 1 side opposite θ y 0 ∞ a sin θ = = hypotenuse r Laws of exponents and roots side adjacent to θ x am 1 cos θ = = aman = am+n, = am−n, a−m = hypotenuse r an am side opposite θ y tan θ = = (ab)m = ambm, (am)n = amn side adjacent to θ x side adjacent to θ x cot θ = = m a √a side opposite θ y am/n = √n am = √n a , √ab = √a√b, = b √ hypotenuse r r b sec θ = = Factoring side adjacent to θ x hypotenuse r a2 b2 =(a b)(a + b), a3 b3 =(a b)(a2 ab + b2) csc θ = = − − ± ± ∓ side opposite θ y 2 2 2 3 3 2 2 3 sin θ 1 (a b) = a 2ab + b , (a b) = a 3a b + 3ab b tan θ = , cot θ = ± ± ± n ± ± cos θ tan θ (a + b)n = an + nan−1b + + an−kbk + + 3abn−1 + bn ··· k ··· 1 1 csc θ = , sec θ = n n(n 1) (n k + 1) sin θ cos θ where = − ··· − k 1 2 3 k sin( θ)= sin θ, cos( θ) = cos θ, tan( θ)= tan θ − − − − − · · ····· (sin θ)2 = sin2 θ, (cos θ)2 = cos2 θ The quadratic formula The following relationships apply to any triangle, (not nec- If a = 0 and ax2 + bx + c = 0, then 6 essarily a right triangle), with angles α, β, and γ; and with b √b2 4ac opposite sides a, b, and c, respectively: x = − ± − 2a α + β + γ = 180◦ a b c Linear equations law of sines = = sin α sin β sin γ A linear equation has the general form y = mx + b, where 2 2 2 law of cosines a = b + c 2bc cos α m and b are constants. The constant b, is the y-intercept. − The constant m is equal to the slope of the straight line: b2 = a2 + c2 2ac cos β, c2 = a2 + b2 2ab cos γ − − y y ∆y m = 2 − 1 = Trigonometric identities x x ∆x 2 2 2 2 2 2 2 − 1 sin θ+cos θ = 1, sec θ = 1+tan θ, csc θ = 1+cot θ Logarithms sin2θ = 2sin θ cos θ, cos 2θ = cos2 θ sin2 θ − a m 1 + cos 2θ 1 cos 2θ ln(ab)=ln a+ln b, ln = ln a ln b, ln(a )= m ln a cos2 θ = , sin2 θ = − b − 2 2 y ln1=0, ln e = 1, For y = ln x x = e sin(α β) = sin α cos β cos α sin β ⇒ ± ± Geometry cos(α β) = cos α cos β sin α sin β ± ∓ In the following: A = area, B = area of base, C = tan α tan β tan(α β)= ± circumference, S = surface area, V = volume, r = radius. ± 1 tan α tan β ∓ Triangle of base a and altitude h: A = 1 ah sin α sin β = 2sin 1 (α β)cos 1 (α β) 2 ± 2 ± 2 ∓ Circle: C = 2πr, A = πr2 cos α + cos β = 2cos 1 (α + β)cos 1 (α β) 2 2 − 1 1 Sphere: S = 4πr2, V = 4 πr3 cos α cos β = 2sin (α + β)sin (α β) 3 − − 2 2 − 2 2 sin α sin β = 1 cos(α β) 1 cos(α + β) Cylinder of height h: A = 2πr + 2πrh, V = πr h 2 − − 2 1 1 2 1 2 Cone cos α cos β = 2 cos(α β)+ 2 cos(α + β) of lateral length s: A = πr + πrs, V = 3 πr h − sin α cos β = 1 sin(α β)+ 1 sin(α + β) Pythagorean theorem with a right angle between a and 2 − 2 2 2 2 π π b, and hypotenuse c: a + b = c . sin α = cos α, cos α = sin α − 2 − − 2 The distance d between two points having coordinates π π (x ,y )and (x ,y ): d = (x x )+(y y ) sin α + = cos α, cos α + = sin α 1 1 2 2 2 − 1 2 − 1 2 2 − p 11 7.2 Derivatives & Integrals 7.3 Table of Integrals Derivatives: Let a and n be constants, and let f and g xn+1 be functions of x; one finds the following derivatives: xn dx = (n = 1) da d(af) df d(xn) n + 1 6 − = 0, = a , = nxn−1 Z dx dx dx dx udv = uv vdu d(eax) d ln ax a − = aeax, = Z Z dx dx x sin xdx = cos x d sin ax d cos ax − = a cos ax, = a sin ax Z dx dx − d(f + g) df dg d(fg) dg df cos xdx = sin x = + , = f + g Z dx dx dx dx dx dx 1 d f 1 df f dg d2f d df dx = ln x = , = x dx g g dx − g2 dx dx2 dx dx Z ln xdx = x ln n x − The chain rule: for a function f = f(g), where g = g(x): Z 1 df df dg eax dx = eax = a dx dg dx Z eax Integrals: The indefinite integral (or antiderivative) xeax dx = (ax 1) a2 − of f(x), denoted by f(x) dx, is defined as, Z x2 2x 2 x2eax dx = eax + If dy/dx = Rf(x) then y = f(x) dx a − a2 a3 Z Z m m! xm−k The following are some properties of indefinite integrals: xmeax dx = eax ( 1)k · − (m k)! ak+1 af(x) dx = a f(x) dx, dx = x + C Z kX=0 − · 1 x Z Z Z x sin axdx = sin ax cos ax a2 − a [f(x)+ g(x)] dx = f(x) dx + g(x) dx Z x 1 Z Z Z sin2 axdx = sin(2ax) definite integrals, 2 − 4a Z b x2 x 1 f(x) dx = y(b) y(a) where y(x)= f(x) dx x sin2 axdx = sin(2ax) cos(2ax) − 4 − 4a − 8a2 Za Z Z b a c b c x3 x2 1 x f(x) dx = f(x) dx, f(x) dx = f(x) dx+ f(x) dx x2 sin2 axdx = sin(2ax) cos(2ax) a − b a a b 6 − 4a − 8a3 − 4a2 Z Z Z Z Z Z sin[(a b)x] sin[(a + b)x] Partial Derivatives: If f is a function of x and y then sin ax sin bxdx = − , a2 = b2 when x and y change by dx and dy, respectively, f changes 2(a b) − 2(a + b) 6 Z − by sin[(a b)x] sin[(a + b)x] ∂f ∂f cos ax cos bxdx = − + , a2 = b2 df = dx + dy 2(a b) 2(a + b) 6 ∂x y ∂y x Z − cos[(a + b)x] cos[(a b)x] sin ax cos bxdx = − , a2 = b2 where df is the total differential of f(x,y). The total − 2(a + b) − 2(a b) 6 differential of f = f(x,y,z) is Z − ∞ 1 ∂f ∂f ∂f e−ax dx = (a> 0) df = dx + dy + dz a ∂x ∂y ∂z Z0 yz xz xy ∞ 1 π 1/2 x1/2e−ax dx = Partial derivatives may be taken in any order: 2a a Z0 ∂2f ∂2f ∞ n! = xne−ax dx = (a> 0, n positive integer) ∂x∂y ∂y∂x n+1 0 a Z ∞ In the following, z is a variabale on which x and y depend: 2 1 π 1/2 e−ax dx = 0 2 a ∂f ∂f ∂f ∂y Z ∞ = + −ax2 1 ∂x z ∂x y ∂y x ∂x z xe dx = 0 2a Z ∞ ∂x 1 2 1 π 1/2 = (the inverter) x2e−ax dx = 4 a3 ∂y z (∂y/∂x)z 0 Z ∞ 2 1 ∂x ∂x ∂z 3 −ax x e dx = 2 = (the permuter) 0 2a ∂y z − ∂z y ∂y x Z ∞ 2 3 π 1/2 x4e−ax dx = ∂x ∂y ∂z 5 = 1 (cyclic rule) 0 8 a ∂y ∂z ∂x − Z ∞ ∞ z x y n −ax2 n −ax2 ∂g ∂h x e dx = 2 x e dx (n = 0 or even) df = g(x,y) dx + h(x,y) dy is exact if = −∞ 0 ∂y ∂x Z Z x y Exact (or total) differentials are path-independent. 12 7.4 Power Series 7.5 Spherical Polar Coordinates A power series has the form of an infinite polynomial: spherical coordinates (r, θ, and φ) are convenient for ∞ a + a x + a x2 + a x3 + = a xn (81) systems with spherical symmetry. 0 1 2 3 ··· n θ n=0 z the numbers a ,a ,a , are the coefficientsX of the se- 0 1 2 r ries. A power series can··· be convergent or divergent. One important test of convergence is the d’Alembert ratio test: If y an+1 −1 lim = R (82) x φ n→∞ a n Figure 3: Representation of a spherical coordinate system. the series converges for R