Quantum Chemistry a Concise Reference of Formulae, Concepts & Data

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 Hückel 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 Schrödinger 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 Schrödinger 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- Schrödinger 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 Schrödinger equation for the quantum mechanical The Schrödinger 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 Schrödinger 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 Schrödinger 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 Schödinger Equation [Lˆz, Lˆx]= i~Lˆy [Lˆ , Lˆz]=0 (47) The Schödinger 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 Schödinger 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 designated 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 Hückel MO Theory 6.1 Molecular Orbital Theory Hückel 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 Hückel 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 letter 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 symmetry 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 Vêe 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 Hê 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 î = 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

13 When complex numbers are multiplied, the two quanti- A B =(A B )i +(A B )j +(A B )k (100) ± x ± x y ± y z ± z ties are multiplied as binomials and i2 is replaced by 1. There are two ways to form the product of two vectors. − The complex conjugate of z which is denoted by z∗, is The scalar product yields a scalar quantity (just a num- obtained by changing i to i. If z is a complex number ber), and the vector product yields a vector. The scalar − z = x + iy its complex conjugate is z∗ = x iy product of A and B is defined by − A B = A B cos θ (101) · | || | The product of a complex number and its complex conju- where θ is the angle between A and B. This is often gate is a real number. The square root of zz∗ quantity is referred to as the dot product. The scalar product is com- referred to as the absolute value of z and is represented mutative: A B = B A. The dot product of unit vectors · · by z i, j, and k are ∗ 1/2 2 2 1/2 | | z =(zz ) = x + y (92) i i = j j = k k = 1 1 cos 0◦ = 1 (102) | | · · · | || | A complex number can be represented as a point in a plot i j = j i = i k = k i = j k = k j = 1 1 cos 90◦ = 0 of Im(z) versus Re(z), as shown in Fig 4. The plane of this · · · · · · | || | (103) figure is referred to as the complex plane. If we draw a vec- When A and B are expressed in terms of components, it tor r from the origin of this figure to the point z =(x,y), can be shown that then the length of the vector, r =(x2 + y2)1/2, is z , the A B = A B + A B + A B (104) | | · x x y y z z magnitude of the absolute value of z. The vector product of A and B is defined by Im A B = A B c sin θ (105) × | || | where θ is the angle between A and B and c is a unit x=r cos θ z=x+iy vector perpendicular to the plane formed by A and B. y The direction of c is given by the right-hand rule: If the r y=r sinθ fingers of your right hand move from A to B, then c is in θ the direction of your thumb. This is often referred to as Re the cross product. The cross product is not commutative x because A B = B A. The cross product of unit Figure 4: Representation of a complex number z = x + iy. vectors i, j,× and k are− × i i = j j = k k = 1 1 c sin0◦ = 0 Complex numbers in their polar forms (see Fig 4): × × × | || | i j = j i = 1 1 k sin 90◦ = k x = r cos θ and y = r sin θ (93) × − × | || | j k = k j = 1 1 i sin 90◦ = i × − × | || | z = r cos θ + ir sin θ = r(cos θ + i sin θ) (94) k i = i k = 1 1 j sin 90◦ = j × − × | || | The series expansions of ex, cos x, and sin x can be used When A and B are expressed in terms of components, it to derive Euler’s formula: can be shown that A B =(AyBz AzBy)i+(AzBx AxBz)j+(AxBy AyBx)k e±iθ = cos θ i sin θ (95) × − − − ± This equation can be expressed as a determinant: Thus, we can write sin and cos functions as i j k A B = A A A (106) eiθ + e−iθ eiθ e−iθ × x y z cos θ = sin θ = − (96) Bx By Bz

2 2i The following operator can be used in diﬀerent ways:

∂ ∂ ∂ = i + j + k (107) ∇ ∂x ∂y ∂z 7.7 Vectors y,z x,z x,y A vector quantity has direction as well as magnitude. A 1. If a function f is a function of x, y, and z, then f vector A in a Cartesian coordinate system can be repre- ∇ sented by (gradient of f or “grad f”) is a vector: A = A i + A j + A k (97) x y z ∂f ∂f ∂f where i, j, and k are vectors of unit length, called unit f = i + j + k (108) ∇ ∂x ∂y ∂z vectors, that point along the x, y, and z axes of the coordinate system. The quantities A , A , and A are referred 2. The scalar product of with a vector v yields the x y z ∇ to as components of A; they can be positive or negative. divergence (“div”) of that vector: It follows from the Pythagorean theorem that the length ∂vx ∂vy ∂vz of A is given by v = + + (109) ∇· ∂x ∂y ∂z 2 2 2 1/2 A = A =(Ax + Ay + Az) (98) | | 3. The vector product of with a vector v yields the Two vectors can be added together to get a new vector. curl of the vector: ∇ Vector addition is commutative, ∂v ∂v ∂v ∂v v = curl v = i z y +j x z A + B = B + A = C (99) ∇× ∂y − ∂z ∂z − ∂x When vectors are added or subtracted, their components ∂v ∂v + k y x (110) in the three directions add subtract separately, ∂x − ∂y

14 7.8 Determinants 7.9 Simultaneous Linear Equations A determinant is a square array of n2 elements. The num- Consider the following system of n linear equations in n ber n is the order of the determinant. The value of a deter- unknowns x1, x2, ..., xn: minant is deﬁned as a certain sum of products of subsets a11x1 + a12x2+ + a1nxn = b1 ··· of the elements. For determinants of order 2 and 3, a21x1 + a22x2+ + a2nxn = b2 .··· a11 a12 . (111) = a11a22 a12a21 an1x1 + an2x2+ + annxn = bn a21 a22 − ··· If at least one of the b’s is not zero, we have a system a11 a12 a13 a a a a a a of inhomogeneous linear equations. Let det(aij ) be the a a a = a 22 23 a 21 23 +a 21 22 21 22 23 11 a a − 12 a a 13 a a determinant of the coeﬃcients of the unknowns in (111). a a a 32 33 31 33 31 32 31 32 33 Cramer’s rule states that xk(k = 1, 2,...,n) is given by

The (n 1)-order determinant obtained by deleting the a a a b a a − 11 12 1,k−1 1 1,k+1 1n ith row and the jth column of the nth-order determinant a a ··· a b a ··· a 21 22 ··· 2,k−1 2 2,k+1 ··· 2n is called the minor Mij of the element aij . The cofac- i+j ························ tor Cij of aij is deﬁned as: Cij = ( 1) Mij. Thus the an1 an2 an,k−1 bn an,k+1 ann − x = ··· ··· expansion of a third-order determinant is k det(aij ) a11 a12 a13 If all the b’s in (111) are zero, we have a system of lin- a a a = a C + a C + a C 21 22 23 11 11 12 12 13 13 ear homogeneous equations. There is a nontrivial solution a a a 31 32 33 only if det(aij ) = 0. The solution to the linear homoge-

Any determinant can be expanded using the elements of neous equations will contain an arbitrary constant, and we any row or column and the corresponding cofactors. For cannot determine a unique value for each unknown. We an nth-order determinant, therefore assign an arbitrary value to any one of the un- n n knowns, say xn = c, then we transfer the last term in each det(aij )= aklCkl = alkClk,k =1or2or ... or n of the equations to the right side to get: l=1 l=1 a11x1 + a12x2+ + a1,n−1xn−1 = a1nc Some theoremsX on determinantsX are as follows: ··· − a21x1 + a22x2+ + a2,n−1xn−1 = a2nc ··· . − I. If every element of a row (or column) of a determi- . (112) an1x1 + an2x2+ + an,n−1xn−1 = annc nant is zero, the value of the determinant is zero. ··· − We now have n equations in n 1 unknowns. We discard II. Interchanging any two rows (or columns) multiplies − the value of a determinant by 1. any one of the equations of (112), say the last one. This − III. If any two rows (or columns) of a determinant are gives a system of n 1 linear inhomogeneous equations in n 1 unknowns. We− could then apply Cramer’s rule to identical, the determinant has the value zero. − IV. Multiplication of each element of any one row (or solve for x1, x2, ..., xn−1. any one column) by some constant k multiplies the value of the determinant by k. 7.10 Matrices V. Addition to each element of one row of the same An (m n) matrix A is an ordered set of mn elements × constant multiple of the corresponding element of aij (i = 1, 2, 3,...,m; j = 1, 2, 3,...,n) arranged in a another row leaves the value of the determinant un- rectangular array of m rows and n columns, changed. This theorem also applies to the addition a11 a12 ... a1n of a multiple of one column to another column. a21 a22 ... a2n A =  . . .  (113) VI. The interchange of all corresponding rows and . . . columns leaves the value of the determinant un-    am1 am2 ... amn  changed. (This interchange means that column one If m = n, the array is a square matrix of order n. If m = 1, becomes row one, column two becomes row two, etc.) we have a row matrix. If n = 1, we have a column matrix. The diagonal of a determinant that runs from the top left Matrix Algebra: (A, B, and C have the same m n): × to the lower right is the principal diagonal. A diagonal de- A = B; aij = bij terminant is a determinant all of whose elements are zero A + B = B + A = C; cij = aij + bij except those on the principal diagonal: A B = C; c = a b − ij ij − ij a11 0 0 0 ··· kA = C; cij = kaij 0 a22 0 0 ··· n Multiplication of an m n matrix A and an n p matrix B 0 0 a33 0 n× × = a11a22a33 ann = ann . . . ··· . ··· i = 1, 2,...,m . i=1 AB = C; c = a b , (114) ...... Y ij ik kj j = 1, 2,...,p 0 0 0 a k=1 nn C has dimensions m Xp. Two matrices may be multiplied ··· A determinant whose only nonzero elements occur in only if they are conformable× (the no. of columns of the square blocks centered about the principal diagonal is ﬁrst equals the no. of rows of the second). Matrix multipli- in block-diagonal form. A block-diagonal determinant is cation is associative: A(BC)=(AB)C; and distributive: equal to the product of the determinants of the blocks. A(B + C)=AB + AC; but need not to be commutative.

15 The Zero Matrix: For the zero (or null) matrix 0, all 7.11 Eigenvalues and Eigenvectors the matrix elements are zero, and we have 0+A = A+0 = The set of linear inhomogeneous equations (111) can be A. The product of an m n matrix A with a conformable written as the matrix equation n p zero matrix is the ×m p zero matrix: A0 = 0. × × a11 a12 ... a1n x1 b1 The Transpose: The transpose AT (or A˜ ) of A is the a21 a22 ... a2n x2 b2  . . . .   .  =  .  (118) matrix formed by interchanging rows and columns of A, ...... and aT = a . For a square matrix, AT is found by       mn nm  an1 an2 ... ann   xn   bn  reflecting the elements about the principal diagonal.       Ax = b (119) The Conjugate: The complex conjugate A∗ ofAisformed If det A = 0, the matrix A is said to be nonsingular. If by taking the complex conjugate of each element of A. A is nonsingular,6 the solution is obtained by multiplying each side of (119) by A−1 on the left, for then The Transpose Conjugate: The transpose conjugate −1 (or adjoint) A† of A is formed by taking the transpose x = A b (120) ∗ † ∗ T † ∗ In the case that b = λx, (119) is an eigenvalue equation: of A ; thus A = (A ) and amn = anm. We also have (AB)† = B†A†, (ABC)† = C†B†A†, etc. Ax = λx (121) where A is a square matrix, c is a column vector with Scalar Properties of Square Matrices: (1) The de- at least one nonzero element, and λ is a scalar, then c is terminant of the matrix A, denoted det A or A , is the said to be an eigenvector (or characteristic vector) of A determinant whose elements are the same as the| elements| and λ is an eigenvalue (or characteristic value) of A. An of A. (2) The trace of the matrix A, denoted tr A, is the important special case in computaional quantum chem- sum of its diagonal elements: tr A = i aii. istry is the secular equation (60) where the functions fk n Diagonal Matrix: A square matrixP is diagonal if all its in the linear variation function φ = i=1 cifi are made ∗ off-diagonal elements are zero: aij = aiiδij. to be orthonormal, then Sij = fi fj dτ = δij, and (121) becomes P Unit Matrix: A unit or identity matrix I (or 1) has Hc = WRc (122) elements (I)ij = δij , and IA=AI=A for any matrix A. where H is the square matrix whose elements are Hij = i Hˆ j and c is the column vector of coefficients c . In Inverse Matrix: The inverse of a square matrix A, de- k h(122| ),| iH is a known matrix and c and W are unknowns to noted by A−1, is a matrix such that −1 −1 be solved for. Let the n eigenvalues and the corresponding A A = AA = I (115) (1) eigenvectors of H denoted by W1, W2, ..., Wn and c , A−1 exists if and only if det A = 0. We also have c(2), ..., c(n), so that 6 (AB)−1 = B−1A−1, (ABC)−1 = C−1B−1A−1, etc. (i) (i) Hc = Wic , i = 1, 2,...,n (123) (i) (i) Orthogonal Matrix: An orthogonal matrix is a square where c is a column vector whose elements are c1 ,..., (i) matrix whose inverse is equal to its transpose: cn and the basis functions fi are orthonormal. Further- A−1 = AT (116) more, let C be the square matrix whose columns are the Unitary Matrix: A unitary matrix is one whose inverse eigenvectors of H, and let W be the diagonal matrix whose is equal to its conjugate transpose: diagonal elements are the eigenvalues of H: −1 † A = A (117) (1) (2) (n) Unitary matrices are denoted by U, hence U−1 = U† or c1 c1 ... c1 W1 0 ... 0 † † T (1) (2) (n) U U = I. A real unitary matrix is orthogonal: U = U .  c2 c2 ... c2  0 W2 ... 0 C = W =   ...... Symmetric Matrix: A square matrix is symmetric if all  . . . .  . . . .  (1) (2) (n)    its elements satisfy a = a , i.e. if it is equal to its  c c ... c   0 0 ...Wn  mn nm  n n n    transpose A = AT. The elements of a symmetric matrix The set of n eigenvalue equations (123) can be written as are symmetric about the principal diagonal. the single equation: HC = CW (124) Hermitian Matrix: A square matrix is Hermitian or Provided C has an inverse, we have self-adjoint, if all its elements satisfy a = a∗ , i.e. if it mn nm C−1HC = W (125) is equal to its transpose conjugate A = A†. A real Her- mitian matrix is symmetric. The diagonal elements of a If the columns of C are orthonormal, then C is a unitary ∗ matrix. The eigenvectors of a Hermitian matrix can be Hermitian matrix must be real: amm = amm. chosen to be orthonormal, therefore Eq. (125) becomes Some properties of matrices are shown in the Table below. C†HC = W (126) Matrix Notation Definition/Property For the common case that H is real as well as Hermitian Unit I IA = AI = A Eq. (126) becomes T T T Transpose A amn = anm C HC = W (127) ∗ Complex Conjugate A To find the eigenvalues and eigenvectors of a Hermitian Transpose Conjugate A† (A∗)T −1 −1 −1 matrix of order n, we perform matrix diagonalization: We Inverse A A A = AA = I search for a unitary matrix C such that C†HC is a di- Orthogonal A−1 = AT agonal matrix. The diagonal elements of C†HC are the Unitary U A−1 = A† eigenvalues of H, and the columns of C are the orthonor- Symmetric A = AT A A† mal eigenvectors of H. Hermitian = 16 8 Classical Physics The total mechanical energy Emech is defined as Emech = K + V . If only conservative forces act, Emech 8.1 Classical Mechanics remains constant, i.e. (K1 + V1 = K2 + V2). This is the Newton’s second law of motion: law of conservation of mechanical energy. F = ma (128) The kinetic energy of an n-particle system is the sum of the kinetic energies of the individual particles: where m is mass, F is the vector sum of all forces acting on n 1 it at some instant of time, and a is the acceleration. The K = K + K + + K = m v2 (140) 1 2 ··· n 2 i i particle’s velocity v is the instantaneous rate of change i=1 X of its position vector r with respect to time: The potential energy V of a system of particles is the sum dr v (129) of contributions due to pairwise interactions between par- ≡ dt ticles. Let Vij be the contribution to V due to the forces The magnitude (length) of the vector v is the particle’s acting between particles i and j, then speed v. The particles acceleration a is the instantaneous V = Vij (141) rate of change of its velocity: i j>i X X dv d2r a = (130) The double sum indicates that we sum over all pairs of i ≡ dt dt2 and j values except those with i j. ≥ Newton’s second law F = ma is equivalent to the three The linear momentum, p, of a particle of mass m is equations Fx = max, Fy = may, Fz = maz where Fx and related to its velocity, v, by ax are the x components of the force and the acceleration. 2 2 Therefore (130) becomes ax = d x/dt , and p = mv (142) 2 2 2 d x d y d z The linear momentum vector points in the direction of Fx = m , Fy = m , Fz = m (131) dt2 dt2 dt2 travel of the particle. In terms of the linear momentum, The infinitesimal amount of work dw done on the body by the total energy of a particle and Newton’s second law of the force F is defined as motion are defined as (in one dimension): p2 dp dw Fx dx (132) E = K + V (x)= + V (x), F = (143) ≡ mech 2m dt where Fx is the component of the force in the direction of the displacement. The work w done by F during displace- The rotational motion of a particle about a central point ment of the particle from x1 to x2 is is described by its angular momentum, L. The angular x2 w = F (x) dx (133) momentum is a vector: its magnitude gives the rate at which a particle circulates and its direction indicates the Zx1 axis of rotation, see Fig. 5. In the special case that F is constant during the displacement, Eq. (133) becomes: w = F (x x ) for F constant. z 2 − 1 The kinetic energy K of a particle is defined as L K 1 mv2 = 1 m(v2 + v2 + v2) (134) ≡ 2 2 x y z The work-energy theorem is defined as y p w = K K = ∆K one particle system (135) r 2 − 1 θ x The potential energy V (x, y, z) is defined as a function of x, y, and z whose partial derivatives satisfy Figure 5: The angular momentum L of a particle of mo- ∂V ∂V ∂V mentum p and position r from a fixed center. = F , = F , = F (136) ∂x − x ∂y − y ∂z − z The particle’s angular momentum L with respect to the The potential energy of an object in the earths gravita- coordinate origin is defined as a vector of length rp sin θ tional field g at hight h is defined as (where θ is the angle between r and p) and direction per- V = mgh (137) pendicular to both r and p, that is L = r p (144) The restoring force of a spring, Hooke’s law, is × The components of L are F = kf x (138) − L = yp zp , L = zp xp , L = xp yp x z − y y x − z z y − x where, x is the displacement from the equilibrium position, and kf is the force constant. Since F = dV/dx, the The speed of a particle rotating in a plane about a fixed force in Eq. (138) corresponds to a potential energy center with frequency of rotation ν is 1 2 V (x)= 2 kf x (139) v = 2πrν = rω (145)

17 where ω is the angular velocity (in radians per second), Electric field E at any point is the the force exerted on and is defined as ω = 2πν. The magnitude of the angular a unit charge at that point. The electric field E at Q2 due momentum, L, is given by to Q1 at a distance r, is the electric force F divided by Q2, or L = Iω (146) F Q1 E = = 2 (154) where I is the moment of inertia. For a point particle Q2 4πǫ0ǫr moving in a circle, the moment of inertia is given by Electric potential φ is the potential energy per unit I = mr2 (147) charge. The electric field E is the negative gradient of an electric potential: E = dφ/dr. The electric potential − Thus, L can be defined as is thus Q v φ = E dr = 1 (155) L = Iω =(mr2) = mvr (148) − 4πǫ ǫr r Z 0 Kinetic energy can be written in terms of momentum for a rotating system as 8.4 Magnetism Iω2 (Iω)2 L2 Magnetic fields are caused by moving charges. If a cur- K = = = (149) 2 2I 2I rent I were flowing through a wire, then the magnetic field is a circular vector mapping out a cylinder around the wire The correspondences between linear and rotating systems and having its center at the wire. The magnitude of the are given in the Table bellow magnetic field strength vector, B, is given by Linear motion Angular motion µ I B = B = 0 (156) Mass (m) Momentofinertia(I) | | 2πr Speed (v) Angularspeed(ω) Momentum (p = mv) Angular momentum (L = Iω) where µ0 is the permeability of vacuum or the magnetic Kinetic energy: Rotational kinetic energy: constant. The vector B is called the magnetic induc- 2 2 2 2 K = mv = p K = Iω = L tion or magnetic flux density, SI unit Tesla (T). The 2 2m 2 2I direction of the vector B is given by the “right-hand rule”.

8.2 The Classical Wave Equation An electrical current I going around in a closed loop circle of area A, induces a linear magnetic effect called a mag- The classical law governing wave motion is the wave equa- netic dipole vector m, the magnitude of which is defined tion ∂2D ∂2D ∂2D 1 ∂2D as + + = (150) m = m = I A (157) ∂x2 ∂y2 ∂z2 v2 ∂t2 | | · 2 where x, y, and z are the coordinates, t the time, v the The unit of the magnetic dipole is A m . The direction velocity of propagation, and D the displacement of the of the vector m is given by the “right-hand rule”. When wave. D is a function of the coordinates x, y, and z and a magnetic dipole m is subjected to a magnetic field B, time t, D(x,y,z,t). The maximum displacement, say of a there is a magnetic potential energy of interaction, Emag: string, from its equilibrium horizontal position is called its Emag = m B = m B cos θ (158) amplitude. If v does not depend on the time, then the dis- − · −| || | placement is the product of a function of the coordinates The magnitude of magnetic dipole moment associated only, ψ(x,y,z), and a periodic function of time, ei2πνt, with a charge Q moving in a circle of radius r with speed where ν is the frequency of the wave, and i = √ 1. Then v is: The current is the charge flow per unit time. − i2πνt Qv Qvr Qrp D = ψ(x,y,z)e (151) m = IA = πr2 = = (159) | | 2πr 2 2m and the value of D at any point varies with a period, t0 = 1/ν. where p is the linear momentum. Since the radius vector r is perpendicular to p, we have D may be a complex function, so that D 2 = D∗D and thus D 2 = ψ∗ψei2πνte−i2πνt = ψ∗ψ =| ψ| 2. Applying Qr p Q mL = × = L (160) this result| | to the mean value theorem we obtain:| | 2m 2m 1 t0 D 2 = ψ∗ψ dt = ψ∗ψ = ψ 2 (152) For an electron, Q = e, the magnetic moment due to its | | t | | − 0 Z0 orbital motion is e mL = − L (161) Thus, the time average of the square of the absolute value 2me of the displacement is equal to the square of the absolute The Bohr magneton, µB, is defined as value of the space-dependent part ψ. e~ µ = (162) B 2m 8.3 Electrostatics e Electrostatic force F between two charges Q and Q so that for an electron, the magnetic dipole can be written 1 2 µ separated by a distance r in a medium of dielectric con- m = B L (163) L − ~ stant ǫ is Q1Q2 −24 −1 F = 2 (153) The Bohr magneton has a value of 9.274 10 J T . 4πǫ0ǫr ×

18 9 Molecular&Spectroscopic Data Colour, frequency, and energy of light Bond Lengths and dissociation energies Color λ ν ν˜ E E 298 298 14 −1 4 −1 −1 Re D0 Re D0 nm 10 s 10 cm eV kJ mol pm kJmol−1 pm kJmol−1 Infrared > 1000 < 3.00 < 1.00 < 1.24 < 120 H2 74.1 435.990 BO 120.4 808.8 Red 700 4.28 1.43 1.77 171 He2 3.8 CO 112.8 1076.5 Orange 620 4.84 1.61 2.00 193 Li2 267.3 110.21 NO 115.1 630.57 Yellow 580 5.17 1.72 2.14 206 Be2 245 11.17* FO 135.8 219.54 Green 530 5.66 1.89 2.34 226 B2 159 297 SiO 151.0 799.6 Blue 470 6.38 2.13 2.64 254 C2 124.2 610 PO 147.6 599.1 Voilet 420 7.14 2.38 2.95 285 N2 109.8 945.33 SO 148.1 517.90 Near UV 300 10.0 3.33 4.15 400 O2 120.7 498.36 ClO 157.0 268.85 Far UV < 200 > 15.00 > 5.00 > 6.20 > 598 F2 141.2 158.78 H–OH 95.8 497.02 Na2 307.9 73.0813 O=CO 115.98 532.2 Absorption characteristics of some groups Groupν ˜max λmax εmax K2 390.5 54.63 C2H6 153.5 (CC) 376.0 104 cm−1 nm dm3 mol−1 cm−1 S2 188.9 378.7 109.4 (CH) 423.0 C=C(π π∗) 6.10 163 1.5 104 Se2 216.6 332.6 C2H4 133.9 (CC) 728.3 → 5.73 174 5.5 × 103 Te2 255.7 339 108.7 (CH) 465.3 C=O(n π∗) 3.7–3.5 270–290 10–20× Cl2 198.8 242.6 C2H2 120.3 (CC) 965 –N=N–→ 2.9 350 15 Br2 228.1 192.8 106.0 (CH) 556.1 > 3.9 < 260 Strong I2 266.6 151.088 C6H12 153.6 (CC) HF 91.7 569.87 111.9 (CH) 399.6 –NO2 3.6 280 10 4.8 210 1.0 104 HCl 127.5 431.62 C6H6 139.9 (CC) × HBr 141.4 366.35 110.1 (CH) 473.1 C6H5– 3.9 255 200 5.0 200 6.3 103 HI 160.9 298.407 Cl–CH3 178.4 356 5.5 180 1.0 × 105 LiH 159.5 238.05 Br–CH3 192.9 297 2+ × [Cu(OH2)6] (aq) 1.2 810 10 NaH 188.7 185.69 I–CH3 213.9 239 2+ [Cu(NH3)4] (aq) 1.7 600 50 −1 ∗ 3 Typical vibrational wavenumbers, ν/˜ cm H2O(n π ) 6.0 167 7.0 10 → × C–H stretch 2850–2960 medium to strong C–H bend 1340–1465 Nuclear spin properties C–C stretch, bend 700–1250 Nuclide Natural magnetic g-value γ NMR C=C stretch 1620–1680 medium (Spin I) abund. moment 107 frequency −1 C C stretch 2100–2260 medium % µ/µN (T s) ν/MHz ≡ · O–H stretch 3590–3650 strong, broad 1n∗ (1/2) 1.9130 3.8260 18.324 29.164 H-bonds 3200–3570 1H (1/2) 99.9844− 2.79285− 5.5857− 26.752 42.576 C=O stretch 1640–1780 strong 1H (1) 0.0156 0.857 44 0.857 44 4.1067 6.536 C N stretch 2215–2275 medium 3 ∗ ≡ H (1/2) 2.978 96 4.2553 20.380 45.414 N–H stretch 3200–3500 medium 10B (3) 19.6 1.8006− 0.6002− 2.875 4.575 C–F stretch 1000–1400 11B (3/2) 80.4 2.6886 1.7923 8.5841 13.663 C–Cl stretch 600–800 strong 13C (1/2) 1.108 0.7024 1.4046 6.7272 10.708 C–Br stretch 500-600 strong 14N (1) 99.635 0.403 56 0.403 56 1.9328 3.078 C–I stretch 500 strong 17O (5/2) 0.037 1.89379 0.7572 3.627 5.774 2− 19 − − − CO3 1410–1450 strong F (1/2) 100 2.628 87 5.2567 25.177 40.077 − 31 NO3 1350–1420 strong P (1/2) 100 1.1316 2.2634 10.840 17.251 − 33 NO2 1230–1250 strong S (3/2) 0.74 0.6438 0.4289 2.054 3.272 2− 35 SO4 1080–1130 strong Cl (3/2) 75.4 0.8219 0.5479 2.624 4.176 Silicates 900–1100 37Cl (3/2) 24.6 0.6841 0.4561 2.184 3.476

Spectroscopic constants of diatomic molecules State Te ωe ωexe Be αe Re NAµ D0 Ei cm−1 cm−1 cm−1 cm−1 cm−1 pm 10−3 kg mol−1 eV eV 79 1 + −4 Br2 Σg 0 325.321 1.077 0.082107 3.187 10 228.10 39.459166 1.9707 10.52 12 1 + × C2 Σg 0 1854.71 13.34 1.8198 0.0176 124.25 6.000000 6.21 12.15 12 1 2 C H Πr 0 2858.5 63.0 14.457 0.534 111.99 0.929741 3.46 10.64 35 1 + −3 Cl2 Σg 0 559.7 2.67 0.2439 1.4 10 198.8 17.484427 2.4794 11.50 12C16O 1Σ+ 0 2169.814 13.288 1.9313 0.017504× 112.832 6.856209 11.09 14.01 1 1 + H2 Σg 0 4401.21 121.34 60.853 3.062 74.144 0.503913 4.4781 15.43 1 + Σu 91700 1358.09 20.888 20.015 1.1845 129.28 1H81Br 1Σ+ 0 2648.98 45.218 8.46488 0.23328 141.443 0.995427 3.758 11.67 1H35Cl 1Σ+ 0 2990.95 52.819 10.5934 0.30718 127.455 0.979593 4.434 12.75 1H127I 1Σ+ 0 2309.01 39.644 6.4264 0.1689 160.916 0.999884 3.054 10.38 127 1 + −4 I2 Σg 0 214.50 0.614 0.03737 1.13 10 266.6 63.452238 1.5424 9.311 39K35Cl 1Σ+ 0 281 1.30 0.12864 7.89 × 10−4 266.665 18.429176 4.34 8.44 14 1 + × N2 Σg 0 2358.57 14.324 1.99824 0.017318 109.769 7.001537 9.759 15.58 3 Πg 59619 1733.39 14.122 1.6375 0.0179 121.26 3 Πu 89136 2047.18 28.445 1.8247 0.0187 114.87 23Na35Cl 1Σ+ 0 366 2.0 0.21806 1.62 10−3 236.08 13.870687 4.23 8.9 16 3 − × O2 Σg 0 1580.19 11.98 1.44563 0.0159 120.752 7.997458 5.115 12.07 1 ∆g 7918.1 1483.5 12.9 1.4264 0.0171 121.56 3 − Σu 49793.3 709.31 10.65 0.8190 0.01206 160.43 16 1 2 O H Πi 0 3737.76 84.811 18.911 0.7242 96.966 0.958087 4.392 12.9

19 10 Other Useful Data

10.1 SI Units & Unit Preﬁxes SI Base Units

Physicalquantity Symbolfor SIunit Symbol quantity forunit Length l metre m Mass m kilogram kg Time t second s Thermodynamic temperature T kelvin K Electric current I ampere A Luminous intensity Iv candela cd Amount of substance n mole mol

Selected SI Derived Units

Physicalquantity Name Symbol Definetion Other of unit form Force newton N kgms−2 Pressure pascal Pa kgm−1 s−2 N m−2 Energy,work,heat joule J kgm2 s−2 N m Electriccharge coulomb C As Electricpotential volt V kgm2 s−3 A−1 J C−1 Electricresistance ohm Ω kgm2 s−3 A−2 V A−1 Electricconductance siemens S kg−1 m−2 s3 A2 Ω−1 Electriccapacitance farad F kg−1 m−2 s4 A2 C V−1 Magneticflux weber Wb kgm2 s−2 A−1 Vs Magneticinductance henry H kgm2 s−2 A−2 V A−1 s Magneticfluxdensity tesla T kgs−2 A−1 V s m−2 Frequency hertz Hz s−1 Activity(radioactive) becquerel Bq s−1

Unit Preﬁxes

a f p n µ mc d kMGTPE atto femto pico nano micro milli centi deci kilo mega giga tera peta exa 10−18 10−15 10−12 10−9 10−6 10−3 10−2 10−1 103 106 109 1012 1015 1018

10.2 Energy Conversion Factors

hartree J eV kcal/mol cm−1 kJ/mol 1 hartree 1.0 4.35974812 10−18 27.2113957 627.509541 219474.625 2625.49992 × 1 J 2.29371049 1017 1.0 6.24150636 1018 1.43932522 1020 5.03411250 1022 6.02213670 1020 × × × × × 1 eV 0.0367493095 1.60217733 10−19 1.0 23.0605423 8065.54093 96.4853090 × 1 kcal/mol 0.00159360127 6.94770014 10−21 0.0433641146 1.0 349.755041 4.184 × 1 cm−1 4.55633538 10−6 1.98644746 10−23 0.000123984245 0.00285914392 1.0 0.0119626582 × × 1 kJ/mol 0.000380879844 1.66054019 10−21 0.0103642721 0.239005736 83.5934612 1.0 ×

10.3 Symbols for Elementary Particles

p, p+ proton µ− negative muon p antiproton d deuteron n neutron t triton n antineutron h(3He2+) helion e, e−, β− electron α (4He2+) α-particle + + e , β positron νe (electron) neutrino γ photon νe electron antineutrino µ+ positive muon

20 10.4 Fundamental Physical Constants Quantity Symbol Value Speed of light in vacuum c 2.99792558 108 ms−1 Elementary charge e 1.602176462× 10−19 C × 23 −1 Avogadro’s constant NA 6.02214199 10 mol × 4 −1 Faraday’s constant F = NAe 9.64853419 10 C mol Gas constant R 8.314472 J K×−1 mol−1 8.314472 10−2 bar dm3 mol−1 K−1 8.20574 ×10−2 atm dm3 mol−1 K−1 1.98719× cal mol−1 K−1 8.314472 107 erg mol−1 K−1 6.23637 ×101 Torr dm3 mol−1 K−1 × −23 −1 Boltzmann’s constant kB = R/NA 1.3806503 10 JK Planck’s constant h 6.62606876× 10−34 Js ~ = h/2π 1.054571596× 10−34 Js × −31 Rest mass of electron me 9.10938188 10 kg × −27 Rest mass of proton mp 1.67262158 10 kg × −27 Rest mass of neutron mn 1.6749286 10 kg × −27 Rest mass of deuteron md 3.3435860 10 kg 12 × −27 Atomic mass constant mu = (1/12)m( C) 1.6605402 10 kg 2 × −12 −1 2 −1 Permittivity of vacuum ǫ0 = 1/µ0c 8.854187817 10 J C m −7 ×2 −2 −1 Permeability of vacuum µ0 4π 10 Js C m × 7 −1 Rydberg constant R∞ 1.097373157 10 m 2 × −18 Hartree energy Eh = e /4πǫ0a0 4.35974434 10 J 2 2 × −11 Bohr radius a0 = 4πǫ0~ /mee 5.291772083 10 m ×−24 −1 Bohr magneton µB = e~/2me 9.274009 10 J T × −27 −1 Nuclear magneton µN = e~/2mp 5.050783 10 J T Electron magnetic moment µ 9.284764 × 10−24 J T−1 e × Electron g value ge 2.002319304 Proton g value gp 5.585695 5 4 3 2 −8 −2 −4 −1 Stefan-Boltzmann constant σ = 2π kB/15h c 5.670373 10 J m K s × 8 −1 −1 Proton gyromagnetic ratio γp 2.67522212 10 s T Fine-structure constant α 7.29735257 × 10−3 Standard acceleration of free fall g 9.80665 m s−×2 Gravitational constant G 6.673 10−11 N m2 kg−2 × 10.5 Atomic Units PhysicalQuantity Symbolforunit Valueina.u. ValueofUnit in SI −11 Length a0 1 5.29177249 10 m × −31 Mass me 1 9.10938188 10 kg Charge e 1 1.602176462× 10−19 C Angular momentum ~ = h/(2π) 1 1.054571596 × 10−34 Js × −18 Energy Eh 1 4.3597482 10 J × 9 3 −2 −2 Coulomb’s constant ke = 1/(4πε0) 1 8.9875518 10 kg m s C × −17 Time ~/Eh 1 2.4188843341 10 s × −6 −1 Velocity a0Eh/~ 1 2.18769142 10 ms × −8 Force Eh/a0 1 8.2387295 10 N × 11 −1 Electric ﬁeld Eh/ea0 1 5.14220624 10 V m × −30 Electric dipole moment ea0 1 8.4783579 10 C m 2 2 × −41 2 2 −1 Electric polarizability e a0/Eh 1 1.648777251 10 C m J Magnetic dipole moment e~/m 1 1.85480308 ×10−23 J T−1 e × 10.6 The Greek Alphabet A, α alpha H, η eta N, ν nu Υ, υ upsilon B, β beta Θ, θ theta Ξ, ξ xi Φ, φ phi Γ, γ gamma I, ι iota Π, π pi X, χ chi ∆, δ delta K, κ kappa P, ρ rho Ψ, ψ psi E, ε epsilon Λ, λ lambda Σ, σ sigma Ω, ω omega Z, ζ zeta M, µ mu T, τ tau

21 10.7 Periodic Table of the Elements 1 18 1A 8A 1 2 H He hydrogen 2 13 14 15 16 17 helium 1.008 2A 3A 4A 5A 6A 7A 4.0026 3 4 5 6 7 8 9 10 Li Be B C N O F Ne lithium beryllium boron carbon nitrogen oxygen fluorine neon 6.941 9.0122 10.811 12.011 14.007 15.999 18.998 20.180 11 12 13 14 15 16 17 18 Na Mg Al Si P S Cl Ar sodium magnesium 3 4 5 6 7 8 9 10 11 12 aluminum silicon phosphorus sulfur chlorine argon 22.990 24.305 3B 4B 5B 6B 7B 8B 8B 8B 1B 2B 26.982 28.085 30.974 32.065 35.453 39.948 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr potassium calcium scandium titanium vanadium chromium manganese iron cobalt nickel copper zinc gallium germanium arsenic selenium bromine krypton 39.098 40.078 44.956 47.867 50.942 51.996 54.938 55.845 58.933 58.693 63.546 65.382 69.723 72.630 74.922 78.971 79.904 83.798 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe rubidium strontium yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon 85.468 87.62 88.906 91.224 92.906 95.95 (98) 101.07 102.91 106.42 107.87 112.41 114.82 118.71 121.76 127.60 126.90 131.29 55 56 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba 57-71 Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn caesium barium hafnium tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon 132.91 137.33 178.49 180.95 183.84 186.21 190.23 192.22 195.08 196.97 200.59 204.38 207.2 208.98 (209) (210) (222) 87 88 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 Fr Ra 89-103 Rf Db Sg Bh Hs Mt Ds Rg Cn Nh Fl Mc Lv Ts Og francium radium rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium roentgenium copernicium nihonium flerovium moscovium livermorium tennessine oganesson (223) (226) (267) (268) (269) (270) (277) (278) (281) (282) (285) (286) (289) (290) (293) (294) (294)

57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 Lanthanides La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu lanthanum cerium praseodymium neodymium promethium samarium europium gadolinium terbium dysprosium holmium erbium thulium ytterbium lutetium 138.91 140.12 140.91 144.24 (145) 150.36 151.96 157.25 158.93 162.50 164.93 167.26 168.93 173.05 174.97 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Actinides Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr actinium thorium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium lawrencium (227) 232.04 231.04 238.03 (237) (244) (243) (247) (247) (251) (252) (257) (258) (259) (266) Notes: 1. This table is based on “IUPAC Periodic Table of the Elements”, dated November 28, 2016. For updates, see www.iupac.org. 2. The upper number in a box is the atomic number, the lower number is the conventional atomic mass. 3. For elements with no stable isotopes, the mass number of the isotope with the longest half-life is in paranthesis. 4. Color code: Blue: gas, Red: liquid, Yellow: artiﬁcially prepared, Gray: Metalloids. 10.8 Atomic Masses for Selected Isotopes Isotope Mass/amu Isotope Mass/amu Isotope Mass/amu Isotope Mass/amu 1H 1.007825 10B 10.012937 18O 17.999165 32S 31.972071 2H 2.014102 11B 11.009305 19F 18.998403 33S 32.971458 3H 3.016049 12C 12.00000 20Ne 19.992440 34S 33.967867 3He 3.016029 13C 13.00335 21Ne 20.993847 35Cl 34.968853 4He 4.002603 14N 14.003074 22Ne 21.991386 37Cl 36.965903 6Li 6.015122 15N 15.00011 23Na 22.989770 79Br 78.918337 7Li 7.016004 16O 15.994915 27Al 26.981538 81Br 80.916291 9Be 9.012182 17O 16.99913 31P 30.973762 127I 126.904473

10.9 Eﬀective Nuclear Charge (Zeﬀ = Z σ) H − He 1s 1 1.6875 Li Be B C N O F Ne 1s 2.6906 3.6848 4.6795 5.6727 6.6651 7.6579 8.6501 9.6421 2s 1.2792 1.9120 2.5762 3.2166 3.8474 4.4916 5.1276 5.7584 2p 2.4214 3.1358 3.8340 4.4532 5.1000 5.7584 Na Mg Al Si P S Cl Ar 1s 10.6259 11.6089 12.5910 13.5745 14.5578 15.5409 16.5239 17.5075 2s 6.5714 7.3920 8.3736 9.0200 9.8250 10.6288 11.4304 12.2304 2p 6.8018 7.8258 8.9634 9.9450 10.9612 11.9770 12.9932 14.0082 3s 2.5074 3.3075 4.1172 4.9032 5.6418 6.3669 7.0683 7.7568 3p 4.0656 4.2852 4.8864 5.4819 6.1161 6.7641