5. Atoms and the periodic table of chemical elements
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1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical elements 6 Diatomic molecules 7 Ten-electron systems from the second row 8 More complicated molecules FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 The one-electron atom H (Z = ), He + (Z = ), Li + (Z = 3),, U 9+ (Z = 9), Electronic Hamilton operator (for point-like clamped nucleus) : Ĥ el = T e + V en = Z e m e κ 0 r (39) Determination of stationary bound states, ie solutions r ψ = ψ(r) with E < 0, to the time-independent Schrödinger equation (with Ĥel from above): (Ĥel E ) ( ψ(r) = Z ) e m e κ 0 r E ψ(r) = 0 (40) Firstly, we remove the fundamental constants by switching to atomic units This reduces the mathematical work to pure numbers, and eliminates quantities which have experimental uncertainties The finite mass of the nucleus can be taken into account by switching from the electron mass me to a reduced mass µ, where µ = m e + m N and m N is the nuclear mass FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
2 Atomic units a Physical quantity Symbol (name) Value in SI units b mass m e (6) 0 3 kg charge e (4) 0 9 C angular momentum, action (8) 0 34 J s el permittivity 4πε 0 κ F m length κ 0 /(m e e ) = /(αm e c) a 0 (bohr) m time /E h = /(α m e c ) s velocity a 0 E h / = αc m s linear momentum /a 0 = αm e c kg m s force E h /a N energy e /(κ 0 a 0 ) = α m e c E h (hartree) J power E h / W charge density e/a C m 3 el current ee h / A el potential E h /e V el capacitance κ 0 a F el resistance /e Ω el field strength (E) E h /(ea 0 ) V m el displacement (D) e/a C m el dipole moment ea C m el quadrupole moment ea C m el polarizability (ea 0 ) /E h C m J magn flux /e Wb magn flux density (B) /(ea 0 ) T magnetizing force (H) ee h /(a 0 ) A m magn dipole moment e /m e = µ B J T a Based on CODATA recommended values 00 ( b The standard deviation uncertainty in the least significant digits is given in parentheses Now the Schrödinger equation reads (Ĥel E ) ψ(r) = ( Z r E ) ψ(r) = 0 (4) which is transformed from cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ), with 0 r <, 0 θ π, and 0 φ π Laplace operator and squared angular momentum operator l (in atomic units) in spherical coordinates: = = r r r r l l = { sin θ l r r r = r θ sin θ θ + sin θ r (4) } φ (43) Separation ansatz for the state function: ψ(r) = ψ(r, θ, φ) = R(r) Y (θ, φ) (44) (this is always possible for central fields, ie V = V (r)) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
3 The angular part is found to be a spherical harmonic, Y (θ, φ) = Y lm (θ, φ), which is an angular momentum eigenfunction, ie a simultaneous eigenfunction of l and l z (in atomic units): l Y lm (θ, φ) = l(l + ) Y lm (θ, φ) (45) l z Y lm (θ, φ) = m Y lm (θ, φ) (46) Orbital angular momentum quantum number l: l = 0,,, 3,, Magnetic quantum number m: l m l Explicit form for the spherical harmonics (with Condon-Shortley phase convention), l m l: Y lm (θ, φ) = Θ lm (θ) Φ m (φ) = ( ) m N lm Pl m (cos θ) e i m φ (47) N lm = l + 4π (l m)! (l + m)!, Y l, m(θ, φ) = ( ) m Y lm (θ, φ) Ω Y lm (θ, φ) Y l m (θ, φ) dω = δ ll δ mm E U Condon (90-974), G H Shortley ( 90) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Notation for single-particle eigenfunctions of angular momentum: l ψ αlm = l(l + ) ψ αlm l z ψ αlm = m ψ αlm l s p d f g h i k l m n Notation for many-particle eigenfunctions of total angular momentum: L z = L Ψ αlm = L(L + ) Ψ αlm L z Ψ αlm = M Ψ αlm n i= l z,i, L = n i= l i, L = L L L S P D F G H I K L M N FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
4 The associated Legendre functions Pl m (cos θ) are related to the Legendre polynomials P l (x) = Pl 0 (x) (x = cos θ) The following relations hold (for l m l, where applicable): Pl m (x) = l l! ( x m/ dl+m ) dx l+m(x ) l (48) P l (x) = d l l l! dx l(x ) l = F ( l, l + ; ; ( x)/) (49) Pl m (x) = ( ) m (l m)! (l + m)! P l m (x) (50) Pl+ m l + (x) = l m + x P l m (x) l + m l m + P l m (x) (5) P m+ l (x) = m x x P m l (x) (l + m)(l m + ) P m l (x) (5) A-M Legendre (75-833) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 The first Legendre polynomials l P l (x) 0 x (3x ) 3 (5x 3 3x) 4 8 (35x 4 30x + 3) 5 8 (63x5 70x 3 + 5x) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
5 The first associated Legendre functions l m P m l (cos θ) l m P m l (cos θ) cos θ 3 + (5 cos3 θ 3 cos θ) 3 sin θ (5 cos θ ) + sin θ 3 8 sin θ (5 cos θ ) sin θ sin θ cos θ 0 (3 cos θ ) 3 8 sin θ cos θ + 3 sin θ cos θ sin 3 θ sin θ cos θ sin3 θ + 3 sin θ 8 sin θ FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 The first spherical harmonics (Condon-Shortley phase convention) l m Y lm (θ, φ) l m Y lm (θ, φ) 0 0 4π π sin3 θ e 3iφ + 3 8π sin θ e iφ π sin θ cos θ e iφ 0 3 4π cos θ π sin θ (5 cos θ ) e iφ 3 8π sin θ e iφ π (5 cos 3 θ 3 cos θ) π sin θ e iφ π sin θ (5 cos θ ) e iφ π sin θ cos θ eiφ π sin θ cos θ e iφ 0 5 4π (3 cos θ ) π sin3 θ e 3iφ 3 5 4π sin θ cos θ e iφ π sin θ e iφ FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
6 Two-point boundary value problem for P (r) = r R(r) (as resulting from the separation ansatz): ( ) d dr + [ E V l(r) ] P (r) = 0 (53) l(l + ) V l (r) = r Z, P (0) = 0, lim r r P (r) = 0 Physically acceptable (ie normalizable) solutions exist only for a discrete set of energies: Quantization due to the boundary conditions Radial functions (eigenfunctions): P nl (r) = r R nl (r) = N nl x l+ L n (l+) r (x) e x/ (54) x = Z n r, N nl = n r! Z n (n + l)!, n r = n l 0 R nl(r) R n 0 l (r) r dr = P nl(r) P n 0 l (r) dr = δ nn FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Energy eigenvalues (in atomic units): E n = E nlm = n (55) These are degenerate for n > (further details below) Principal quantum number n: n =,, 3, This result for E n is also an important hint for the understanding of the stability of matter: E n >, ie the electron does not collapse into the nucleus, despite the singular attractive potential, due to a balance between kinetic and potential energy Generalized Laguerre polynomials: Z L (α) k (x) = (α + ) k k! F ( k; α + ; x) (56) L (α) k (x) = k + α x L (α) k k (x) k + α L (α) k k (x) (k ) (57) E N Laguerre ( ) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
7 The first generalized Laguerre polynomials k 0 L (α) k (x) (α + ) x [ (α + )(α + ) (α + )x + x ] [ (α + )(α + )(α + 3) 3(α + )(α + 3)x + 3(α + 3)x x 3 ] [ (α + )(α + )(α + 3)(α + 4) 4(α + )(α + 3)(α + 4)x + 6(α + 3)(α + 4)x 4(α + 4)x 3 + x 4 ] [ (α + )(α + )(α + 3)(α + 4)(α + 5) 5(α + )(α + 3)(α + 4)(α + 5)x + 0(α + 3)(α + 4)(α + 5)x 0(α + 4)(α + 5)x 3 + 5(α + 5)x 4 x 5 ] FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 The first radial functions of one-electron atoms n l n r P nl (r) (x = Zr/n) s 0 0 Z x e x/ s 0 Z x ( x) e x/ p 0 Z6 x e x/ 3s 3 0 3p 3 3d 3 0 4s p 4 4d 4 4f Z3 3 x (3 3x + x /) e x/ Z4 x (4 x) e x/ 3 Z 3 0 x 3 e x/ Z4 4 x (4 6x + x x 3 /6) e x/ Z60 x (0 5x + x /) e x/ 4 Z 4 70 x 3 (6 x) e x/ 4 Z 5040 x4 e x/ FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
8 Radial functions P nl (r) of the one-electron atom (for n =,, 3, 4) P nl (r) Z / n = (s) P nl (r) Z / n = 3 (3s, 3p, 3d) Zr Zr P nl (r) Z / 04 0 n = (s, p) P nl (r) Z / 04 0 n = 4 (4s, 4p, 4d, 4f) Zr Zr FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Eigenfunctions and energy eigenvalues (in atomic units) of the oneelectron atom: ψ nlm (r) = r P nl(r) Y lm (θ, φ), E nlm = E n = Z n (58) n =,, 3, 4,, l = 0,,, n, m = l, l +,, l nlm n l m = ψ nlm (r) ψ n l m (r) dr = δ nn δ ll δ mm Ground state and corresponding energy: ψ 00 (r) = Z 3 π e Zr = r Z Zr e Zr 4π, E = Z (59) For isovalue plots ie representations of all points r with ψ nlm (r) = c for chosen c R of the eigenfunctions of the one-electron atom, see J Brickmann, M Klöffler, H-U Raab, Chemie in unserer Zeit (978) 3-6 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
9 Degree of degeneracy : m degeneracy: g l = l degeneracy (without spin): g n = l m= l n l=0 = l + (60) g l = n (6) l degeneracy (spin included): g s n = g n = n (6) The value g s n essentially determines the length of the rows ( periods ) in the table of chemical elements (, 8, 8, 3), whereas the value g l = (l + ) determines the block structure of the periodic table (s-, p-, d-, and f-block for l = 0,,, 3, respectively) The degeneracy with respect to l, eq (6), is a special property of the one-electron atom (with point-like nucleus), and is not present in many-electron atoms For example, the s and p states of a one-electron atom are degenerate, ie they have the same energy, but the orbital energies for the s and p orbitals in any state of a many-electron atom are always different FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 The periodic table of the chemical elements (004) H 3 Li Na 9 K 37 Rb 55 Cs 87 Fr 4 Be Mg 0 Ca 38 Sr 56 Ba 88 Ra Sc 39 Y Ti 40 Zr 7 Hf 04 Rf 3 V 4 Nb 73 Ta 05 Db 4 Cr 4 Mo 74 W 06 Sg 5 Mn 43 Tc 75 Re 07 Bh 6 Fe 44 Ru 76 Os 08 Hs 7 Co 45 Rh 77 Ir 09 Mt 8 Ni 46 Pd 78 Pt 0 Ds 9 Cu 47 Ag 79 Au 30 Zn 48 Cd 80 Hg ( ) 5 B 3 Al 3 Ga 49 In 8 Tl 3 6 C 4 Si 3 Ge 50 Sn 8 Pb 4 ( ) 7 N 5 P 33 As 5 Sb 83 Bi 5 8 O 6 S 34 Se 5 Te 84 Po 6 ( ) 9 F 7 Cl 35 Br 53 I 85 At 7 He 0 Ne 8 Ar 36 Kr 54 Xe 86 Rn 8 (?) 57 La 89 Ac 58 Ce 90 Th 59 Pr 9 Pa 60 Nd 9 U 6 Pm 6 Sm 93 Np 94 Pu 63 Eu 95 Am 64 Gd 96 Cm 65 Tb 97 Bk 66 Dy 67 Ho 98 Cf 99 Es 68 Er 00 Fm 69 Tm 0 Md 70 Yb 0 No 7 Lu 03 Lr FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
10 The variation principle at work (I) The ground state s S of the one-electron atom Definition of the system under consideration: Ĥ = T + V, T =, V = V nuc (r) = Z r A set of trial functions : Slater function (ζ > 0): ψ S = N S exp ( ζr) 3 Lorentz function (a > 0): ψ L = N L [ + (ar) ] Gauß function (α > 0): ψ G = N G exp ( αr ) 4 Preuß function (c > 0): ψ P = N P [ + cr ] Refs: C Zener, Phys Rev 36 (930) 5, J C Slater, Phys Rev 36 (930) 57 (Slater function) S F Boys, Proc R Soc London A 00 (950) 54, H Preuß, Z Naturforsch A (956) 83 (Gauß function) H Preuß, Z Naturforsch A 3 (958) 439 (Preuß function) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Trial functions (unnormalized) for the ground state of the one-electron atom f(x) S: exp( x), x = ζr G: exp( x ), x = αr L: /( + x ), x = ar P: /( + x), x = cr x FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
11 Estimate for the ground state energy: E = H = ψ Ĥ ψ = ψ(r) Ĥ ψ(r) dr = T + V T = ψ ψ = + ψ ψ, V = Z r Mathematical preliminaries: The beta function: B(a, b) = Γ(a) Γ(b) Γ(a + b) = B(b, a) B(a, b) = 0 ta ( t) b t a dt = dt (a > 0, b > 0) 0 ( + t) a+b Useful integral formulas: 0 xs exp ( p x n ) dx = n Γ(s/n) p s/n (p > 0, s/n > 0) x s 0 ( + x n ) p dx = B(p s/n, s/n) (s/n > 0, np s > 0) n FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 The Slater function ψ S = N S exp ( ζr), ζ opt =?: r k = N S 4π = r 0 = N S dr 0 rk+ e ζr = NS π ζ 3 N S = ζ 3 /π 4π Γ(k + 3) (ζ) k+3 (k > 3) r k Γ(k + 3) = k+ ζ k T = + N S 4π ( ) d ζ dx 0 dx x e x = N S 4π ζ dx ( 0 x) e x = ζ V = Z ζ E = ζ Z ζ d E = ζ Z = 0 ζ opt = Z dζ ( ) E Z = x x (x = ζ/z) E min = Z FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
12 The Gauß function ψ G = N G exp ( αr ), α opt =?: Γ( k+3 ) r k = NG 4π dr 0 rk+ e αr = NG 4π ( ) = r 0 = NG π 3/ ( α NG = α π r k = Γ( k+3 ) π (α) k/ T = + N G 4π ( ) d dx α 0 dx x e x = N G V = Z 4π α α/π 0 dx ( x ) e x = 3 α E = 3 α α Z π d E dα = 3 Z ( ) E 3 Z = x x π (α) k+3 ) 3/4 (k > 3) πα = 0 α opt = 8Z 9π (x = α/z) E min = 4 3π Z FAQC D Andrae, Theoretical Chemistry, U Bielefeld / The Lorentz function ψ L = N L [ + (ar) ], aopt =?: r k = NL 4π r k+ dr 0 ( + (ar) ) = N L 4π a k+3 B( k, k + 3 ) = r 0 = NL π a 3 N L = a 3 /π r k = π a k B( k, k + 3 ) ( 3 < k < ) T = + N L 4π ( ) d a dx x 0 dx + x = N L 4π a dx ( x ) 0 ( + x ) 4 = 4 a V = Z a/π E = 4 a Z a π d E da = a Z π = 0 a opt = 4Z π ( E Z = x 4 x ) (x = a/z) E min = 4 π π Z FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
13 4 The Preuß function ψ P = N P [ + cr ], copt =?: r k = N P 4π 0 r k+ dr ( + cr) 4 = N P = r 0 = NP 4π 3c 3 N P = 4π B( k, k + 3) ck+3 3c 3 r k = 3 B( k, k + 3) ( 3 < k < ) ck T = + N P ( ) 4π d c dx x 0 dx ( + x) = N P 4π ( x) dx c 0 ( + x) 6 = 5 c V = Z c/ E = 5 c Z c d E = dc 5 c Z = 0 c opt = 5Z 4 ( E Z = x 5 x ) (x = c/z) E min = 5 6 Z 4π FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Variation of the ground state energy E of the one-electron atom with the trial function parameter E Z = f(x) = a x + a x f(x) 0 05 x S: x / x, x = ζ/z G: 3x / /π x, x = α/z L: x /4 x/π, x = a/z P: x /5 x/, x = c/z For optimal choice of the parameter x (ie at the minima), the quantum mechanical virial theorem, V / T =, is fulfilled, and thus E = V / = T in all four cases FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
14 Perturbation theory Time-independent Rayleigh-Schrödinger perturbation theory for nondegenerate states Find a solution for the Schrödinger equation ) Ĥ λ ψ λ = E λ ψ λ (Ĥλ E λ ψλ = 0 (63) with the Hamilton operator Ĥ λ = Ĥ(0) + k= λ k Ĥ(k), (64) where λ is a (natural or artificial) perturbation parameter ( λ < λ max ), and assume that the solutions of the unperturbed problem Ĥ (0) ψ (0) = E (0) ψ (0) (65) are completely known (with all E (0) non-degenerate) Then put E λ = E (0) + k= ε (k) λ k and ψ λ = ψ (0) + l= χ (l) λ l (66) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 into eq (63): 0 = Ĥ(0) E (0) + k= = (Ĥ(0) E (0)) ψ (0) + + m k= (Ĥ(k) ε (k) ) λ k ψ (0) + m= l= λ m { (Ĥ(0) E (0) ) χ (m) (Ĥ(k) ε (k)) χ (m k) + (Ĥ(m) ε (m)) ψ (0) } Thus we obtain from the coefficient of λ m : χ (l) λ l m = 0 : m = : m = : (Ĥ(0) E (0) ) ψ (0) = 0 (Ĥ(0) E (0) ) χ () + (Ĥ() E ()) ψ (0) = 0 (Ĥ(0) E (0)) χ () + (Ĥ() E ()) χ () + (Ĥ() E ()) ψ (0) = 0 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
15 The intermediate normalization condition ψ (0) ψ λ = with ψ (0) ψ (0) = implies orthogonality between ψ (0) and all χ (l) : ψ (0) χ (l) = 0 (for l =,, ) Resulting expressions for ε () and ε () : ε () = ψ (0) Ĥ() ψ (0) (67) ε () = ψ (0) Ĥ() χ () + ψ (0) Ĥ() ψ (0) (68) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 The perturbation theory may be used advantageously to determine (non-variational) approximations to the solutions of eq (63): E λ E (n) = E (0) n + ε (k) λ k k= ψ λ ψ (n) = ψ (0) n + χ (l) λ l l= to obtain exact values for derivatives of the energy E or the state function ψ to various orders in λ at λ = 0, eg: n E λ n = n! ε (n) λ=0 The perturbation theory presented above can be extended to include the case of degenerate states FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
16 The two-electron atom H (Z = ), He (Z = ), Li + (Z = 3),, U 90+ (Z = 9), Hamilton operator (in atomic units): Ĥ el = Z r Z r + r = ĥ + ĥ + r (69) General structure of state functions for two-electron systems : Φ(x, x ) = f(r, r )Θ(σ, σ ) Spin part Θ = SM S : Singlet (S = 0, para-he) or triplet (S =, ortho-he) 00 = (αβ βα)/ = αα 0 = (αβ + βα)/ = ββ as long as the Hamilton operator does not act on the spin of the particles FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 For the spatial part f(r, r ) a suitable choice must be made For S states we can simplify further to f(r, r, r ), or equivalently f(s, t, u) with s = r + r, t = r r, and u = r Variational results for the helium ground state s S a f(s, t, u) E opt /au e ζr e ζr = e ζs 8477 ϕ(r ) ϕ(r ) 867 b e ζr e ηr + e ηr e ζr 8757 c e ζs+cu 8896 e ζs ( + cu) 89 d e ζs+cu cosh (at) 8994 e ( ζs c 0 + c u + c t + c 3 s + c 4 s + c 5 u ) 903 exact 9037 e a E A Hylleraas, Z Phys 54 (99) 347 b C Froese Fischer: The Hartree-Fock method for atoms Wiley, New York, 977 c C Eckart, Phys Rev 36 (930) 878 d W-K Li, J Chem Educ 64 (987) 8 e K Frankowski, C L Pekeris, Phys Rev 46 (966) 46 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
17 The many-electron atom A part of the knowledge of the state functions of the one-electron atom can be transferred to the many-electron atom, if the following assumptions are made : Central field approximation: The electrons in the many-electron atom are assumed to move in an effective central field V eff,l (r), so that the orbitals can be written as ψ(r) = R(r) Y (θ, φ), with Y (θ, φ) = Y lm (θ, φ) Equivalence restriction: The radial parts are assumed to be independent of the magnetic quantum number m: R(r) = R nl (r) The resulting set of radial functions P nl (r) = r R nl (r) has to be determined for every state of the many-electron atom, eg - He ground state (singlet): s S P 0 (r) - He excited states (singlet or triplet): s s,3 S P 0 (r), P 0 (r) - Li ground state (doublet): s s S P 0 (r), P 0 (r) In addition to the approximation of the many-electron state function as a Slater determinant (an antisymmetrized product of spin orbitals), or a linear combination thereof FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Electron configuration of neutral atoms in the ground state (designated as S+ L J ) H s S / He s S 0 3 Li [He] s S / 4 Be [He] s S 0 5 B [He] s p P / 6 C [He] s p 3 P 0 7 N [He] s p 3 4 S 3/ 8 O [He] s p 4 3 P 9 F [He] s p 5 P 3/ 0 Ne [He] s p 6 S 0 Na [Ne] 3s S / Mg [Ne] 3s S 0 3 Al [Ne] 3s 3p P / 4 Si [Ne] 3s 3p 3 P 0 5 P [Ne] 3s 3p 3 4 S 3/ 6 S [Ne] 3s 3p 4 3 P 7 Cl [Ne] 3s 3p 5 P 3/ 8 Ar [Ne] 3s 3p 6 S 0 9 K [Ar] 4s S / 0 Ca [Ar] 4s S 0 Sc [Ar] 3d 4s D 3/ Ti [Ar] 3d 4s 3 F 3 V [Ar] 3d 3 4s 4 F 3/ 4 Cr [Ar] 3d 5 4s 7 S 3 5 Mn [Ar] 3d 5 4s 6 S 5/ 6 Fe [Ar] 3d 6 4s 5 D 4 7 Co [Ar] 3d 7 4s 4 F 9/ 8 Ni [Ar] 3d 8 4s 3 F 4 9 Cu [Ar] 3d 0 4s S / 30 Zn [Ar] 3d 0 4s S 0 3 Ga [Ar] 3d 0 4s 4p P / 3 Ge [Ar] 3d 0 4s 4p 3 P 0 33 As [Ar] 3d 0 4s 4p 3 4 S 3/ 34 Se [Ar] 3d 0 4s 4p 4 3 P 35 Br [Ar] 3d 0 4s 4p 5 P 3/ 36 Kr [Ar] 3d 0 4s 4p 6 S 0 37 Rb [Kr] 5s S / 38 Sr [Kr] 5s S 0 39 Y [Kr] 4d 5s D 3/ 40 Zr [Kr] 4d 5s 3 F 4 Nb [Kr] 4d 4 5s 6 D / 4 Mo [Kr] 4d 5 5s 7 S 3 43 Tc [Kr] 4d 5 5s 6 S 5/ 44 Ru [Kr] 4d 7 5s 5 F 5 45 Rh [Kr] 4d 8 5s 4 F 9/ 46 Pd [Kr] 4d 0 S 0 47 Ag [Kr] 4d 0 5s S / 48 Cd [Kr] 4d 0 5s S 0 49 In [Kr] 4d 0 5s 5p P / 50 Sn [Kr] 4d 0 5s 5p 3 P 0 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
18 5 Sb [Kr] 4d 0 5s 5p 3 4 S 3/ 5 Te [Kr] 4d 0 5s 5p 4 3 P 53 I [Kr] 4d 0 5s 5p 5 P 3/ 54 Xe [Kr] 4d 0 5s 5p 6 S 0 55 Cs [Xe] 6s S / 56 Ba [Xe] 6s S 0 57 La [Xe] 5d 6s D 3/ 58 Ce [Xe] 4f 5d 6s G 4 59 Pr [Xe] 4f 3 6s 4 I 9/ 60 Nd [Xe] 4f 4 6s 5 I 4 6 Pm [Xe] 4f 5 6s 6 H 5/ 6 Sm [Xe] 4f 6 6s 7 F 0 63 Eu [Xe] 4f 7 6s 8 S 7/ 64 Gd [Xe] 4f 7 5d 6s 9 D 65 Tb [Xe] 4f 9 6s 6 H 5/ 66 Dy [Xe] 4f 0 6s 5 I 8 67 Ho [Xe] 4f 6s 4 I 5/ 68 Er [Xe] 4f 6s 3 H 6 69 Tm [Xe] 4f 3 6s F 7/ 70 Yb [Xe] 4f 4 6s S 0 7 Lu [Xe] 4f 4 5d 6s D 3/ 7 Hf [Xe] 4f 4 5d 6s 3 F 73 Ta [Xe] 4f 4 5d 3 6s 4 F 3/ 74 W [Xe] 4f 4 5d 4 6s 5 D 0 75 Re [Xe] 4f 4 5d 5 6s 6 S 5/ 76 Os [Xe] 4f 4 5d 6 6s 5 D 4 77 Ir [Xe] 4f 4 5d 7 6s 4 F 9/ 78 Pt [Xe] 4f 4 5d 9 6s 3 D 3 79 Au [Xe] 4f 4 5d 0 6s S / 80 Hg [Xe] 4f 4 5d 0 6s S 0 8 Tl [Xe] 4f 4 5d 0 6s 6p P / 8 Pb [Xe] 4f 4 5d 0 6s 6p 3 P 0 83 Bi [Xe] 4f 4 5d 0 6s 6p 3 4 S 3/ 84 Po [Xe] 4f 4 5d 0 6s 6p 4 3 P 85 At [Xe] 4f 4 5d 0 6s 6p 5 P 3/ 86 Rn [Xe] 4f 4 5d 0 6s 6p 6 S 0 87 Fr [Rn] 7s S / 88 Ra [Rn] 7s S 0 89 Ac [Rn] 6d 7s D 3/ 90 Th [Rn] 6d 7s 3 F 9 Pa [Rn] 5f 6d 7s 4 K / 9 U [Rn] 5f 3 6d 7s 5 L 6 93 Np [Rn] 5f 4 6d 7s 6 L / 94 Pu [Rn] 5f 6 7s 7 F 0 95 Am [Rn] 5f 7 7s 8 S 7/ 96 Cm [Rn] 5f 7 6d 7s 9 D 97 Bk [Rn] 5f 8 6d 7s 8 H 7/ 98 Cf [Rn] 5f 0 7s 5 I 8 99 Es [Rn] 5f 7s 4 I 5/ 00 Fm [Rn] 5f 7s 3 H 6 0 Md [Rn] 5f 3 7s F 7/ 0 No [Rn] 5f 4 7s S 0 03 Lr [Rn] 5f 4 6d 7s 04 Rf [Rn] 5f 4 6d 7s FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 LS terms for electron configurations p and p 3 p : 3 P (9), D (5), S () ( ) 6 = 5 = p 3 : 4 S (4), D (0), P (6) ( ) 6 3 = 0 = Energy levels of neutral tetravalent atoms from the p-, d-, and f-block (C, Ti, Ce), within an energy range above lowest ground state level: E = ev cm (E h = h c R ) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
19 Energy levels of C I (80 lines of data) IP = 6030 ev Configuration Term J Level (cm-) sp 3P sp D 0963 sp S sp3 5S* sp(p*)3s 3P* sp(p*)3s P* 6988 sp3 3D* sp(p*)6d F* sp(p*)7d F* sp(p*)8d F* sp(p*)9d F* C II (P*</>) Limit C II (P*<3/>) Limit sp(4p)3s 5P sp3 3S* sp3 P* [9878] Source: Atomic Spectra Database, FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Energy levels of Ti I (380 lines of data) IP = 680 ev Configuration Term J Level Lande Leading (cm-) g Percentages d4s a 3F d3(4F)4s a 5F d4s a D d3(D)4s D 3d4s a 3P d3(P)4s 3P d3(P)4s 3P d3(P)4s 3P 3d3(4F)4s b 3F d3s G Source: Atomic Spectra Database, FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
20 Energy levels of Ti I (contd) 3d4s a G d3(G)4s G 3d4 a 5D d4p e 3D d4p h 5D FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Energy levels of Ti I (contd) f 3D f D f G e P Ti II (4F<3/>) Limit 5500 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
21 Energy levels of Ce I (953 lines of data) IP = ev Configuration Term J Level Lande Leading (cm-) g Percentages f5d 6s G* H* 4f5d 6s 3F* D* f(F*)5d(D)6s(D) 3F* * f(F*)5d(3F)6s(4F) 5I* 4f5d 6s 3H* G* f(F*)5d(D)6s(D) 3H* f(F*)5d(D)6s(D) 3H* 4f5d 6s 3G* f(F*)5d(D)6s(D) 3G* f(F*)5d(3F)6s(4F) 5H* 4f(F*) 5d(3F)6s (4F) 5H* f5d6s 3G* (F*) (3F)(4F) 3G* (F*) (3F)(4F) 5I* (F*) (3F)(4F) 5I* 4f5d 6s D* F* Source: Atomic Spectra Database, FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Energy levels of Ce I (contd) 4f(F*) 5d(3F)6s (4F) 5I* f5d6s 3F* f5d6s 3H* (F*) (3F)(4F) 5H* (F*) (3F)(4F) 5H* f(F*) 5d(3F)6s (4F) * G* 37 (F*) (3F)(4F) 5H* 4f5d 6s * * F* 6 3G* 4f5d 6s 3D* f(F*)5d(D)6s(D) 3D* f(F*)5d(D)6s(D) 3D* F* 4f(F*) 5d(3F)6s (4F) * D* 3 (F*) (D)(D) 3P* 4f(F*) 5d(3F)6s (4F) * S* 5 (F*) (D)(D) 3P* 4f(F*) 5d(3F)6s (4F) 3G* (F*) (3F)(4F) 5H* 4f(F*) 5d(3F)6s (4F) * G* 30 (F*) (3F)(4F) 5H* 4f(F*) 5d(3F)6s (4F) * G* 8 4f5d6s 3G* FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
22 Energy levels of Ce I (contd) 4f6s 3H f5d(3G*)6s6p(P*) 3H f5d(3G*)6s6p(P*) 3H f5d(3G*)6s6p(P*) 3H 4f(F*) 5d(3F)6s (4F) 3S* (F*) (3F)(4F) 5D* 4f(F*) 5d(3F)6s (4F) * D* 7 (F*) (3F)(4F) 5G* 4f(F*) 5d(3F)6s (4F) 5G* (F*) (3F)(4F) 5D* (F*) (3F)(4F) 5D* (F*) (3F)(F) G* (F*) (3F)(4F) 3H* (F*) (3F)(4F) 3H* 4f(F*) 5d(3F)6s (4F) * D* (F*) (3F)(4F) 3F* 4f(F*) 5d(3F)6s (4F) * D* 35 4f5d6s 3P* 4f(F*) 5d(3F)6s (4F) * D* 7 (F*) (3F)(4F) 3F* 4f5d 6s * P* 6 4f(F*)5d(3F)6s(4F) 5D* 4f(F*) 5d(3F)6s (4F) 5F* (F*) (3F)(4F) 5D* (F*) (3F)(4F) 5G* (F*) (3F)(4F) 5D* (F*) (3F)(4F) 5D* (F*) (3P)(4P) 5F* FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Energy levels of Ce I (contd) 4f5d 6s * P* 6 4f(F*)5d(D)6s(D) 3P* 4f(F*) 5d(3F)6s (F) * H* 7 (F*)(3F)(F) G* 4f5d 6s * F* 7 3D* 4f(F*) 5d(3F)6s (4F) * I* 8 (F*)(3F)(F) 3I* 4f(F*) 5d(3F)6s (4F) * F* 8 (F*)(3F)(4F) 5G* 4f(F*) 5d(3F)6s (4F) * F* 6 (F*)(3F)(4F) 5F* 4f(F*) 5d(3F)6s (F) * G* (F*)(3F)(4F) 3H* 4f(F*) 5d(3F)6s (4F) * I* 9 (F*)(G)(G) 3I* 4f(F*) 5d(3F)6s (4F)? * I* 4 (F*)(3F)(4F) 5G* 4f(F*) 5d(3F)6s (F)? * H* (F*)(3F)(F) 3I* 4f(F*) 5d(D)6s (D) * P* 6 (F*)(3F)(F) 3D* 4f(F*) 5d(3F)6s (4F) * F* 8 (F*)(3F)(4F) 5D* FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
23 Energy levels of Ce I (contd) 4f(F*) 5d(3P)6s (4P) 5G* (F*)(3F)(4F) 5G* (F*)(3P)(4P) 3G* (F*)(3F)(F) 3H* (F*)(3P)(4P) 3G* (F*)(3F)(F) 3H* 4f(F*) 5d(3F)6s (4F) 5P* (F*)(3F)(4F) 5S* (F*)(3F)(4F) 3D* (F*)(3F)(4F) 3P* 4f6s 3F f5d(3F*)6s6p(P*) 3F f5d(3F*)6s6p(P*) 3F G 4f(F*) 5d(3F)6s (F) S* f5d6s 3P* 4f(F*) 5d(3F)6s (4F) * P* 6 (F*)(3F)(4F) 5S* 4f(F*) 5d(3F)6s (F) * I* 9 4f5d6s H* 4f(F*) 5d(3F)6s (F) * H* 9 (F*)(3P)(4P) 5G* 4f(F*) 5d(3F)6s (4F) 3I* (F*)(G)(G) 3I* 4f(F*) 5d(3F)6s (4F) * G* 5 (F*)(3F)(4F) 3H* FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 Energy levels of Ce I (contd) 4f(F*) 5d(3F)6s (4F) * P* (F*)(3F)(4F) 5P* 4f(F*) 5d(3F)6s (F) * F* 4 4f5d6s F* 4f(F*) 5d(3F)6s (F) * H* 7 (F*)(3P)(4P) 5G* 4f(F*) 5d(3F)6s (4F) * P* 3 (F*)(3F)(F) S* 4f(F*) 5d(3P)6s (4P) 3D* (F*)(3F)(4F) 5P* 4f(F*) 5d(3F)6s (F) * D* (F*)(3P)(P) 3D* 4f(F*) 5d(3F)6s (F) 3I* (F*)(G)(G) 3I* (F*)(G)(G) 3I* 4f(F*) 5d(3F)6s (4F) * P* 7 (F*)(3F)(F) 3D* 4f(F*) 5d(3P)6s (4P) 3D* (F*)(3F)(4F) 5S* 4f(F*) 5d(3F)6s (F) 3F* (F*)(3P)(P) 3F* 4f(F*) 5d(3F)6s (F)? 3G* (F*)(3P)(P) 3G* FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
24 Energy levels of Ce I (contd) * * 4, * * * * * ? * * * Ce II (4H*<7/>) Limit 4467 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5 H atom in a weak homogeneous electric field (Stark effect) perturbation theory FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 48-5
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