Section 4-4
Section 4-4 Atomic Units
of the set of associated Laguerre polynomials. Like the Legendre functions, these are mathematically well characterized. A few of the low-index associated Laguerre polynomials are L1(ρ) = 1, L1(ρ) = 2ρ − 4, 1
2
(4-47)
L1(ρ) = −3ρ2 + 18ρ − 18, L3(ρ) = −6
3
3
4-4 Atomic Units
It is convenient to define a system of units that is more natural for working with atoms and molecules. The commonly accepted system of atomic units for some important quantities is summarized in Table 4-1. [Note: the symbol ¯h (“h-cross or h-bar”) is often used in place of h/2π .] Additional data on values of physical quantities, units, and conversion factors can be found in Appendix 10.
In terms of these units, Schr¨odinger’s equation and its resulting eigenfunctions and eigenvalues for the hydrogenlike ion become much simpler to write down. Thus, the TABLE 4-1 Atomic Units
Values of some atomic properties in atomic units
Quantity
Atomic unit in cgs or other units (a.u.)
Mass me = 9.109534 × 10−28 g Mass of electron = 1 a.u.
Length
a0 = 4πε0 ¯h2/mee2
Most probable distance of 1s = 0.52917706 × 10−10 m electron from nucleus of H ( = 1 bohr) atom = 1 a.u
Time τ0 = a0 ¯h/e2 Time for 1s electron in H = 2.4189 × 10−17 s atom to travel one bohr = 1 a.u.
Charge e = 4.803242 × 10−10 esu Charge of electron = −1 a.u.
= 1.6021892 ×10−19 coulomb
Energy e2/4π ε0a0 = 4.359814 × 10−18 J
Total energy of 1s electron in ( = 27.21161 eV ≡ 1 hartree) H atom = −1/2 a.u.
Angular
¯h= h/2π
Angular momentum for momentum
= 1.0545887 × 10−34 J s particle in ring = 0, 1, 2, . . . a.u.
Electric field e/a2 = 5.1423 × 109 V/cm Electric field strength at
0
strength distance of 1 bohr from proton = 1 a.u.
Chapter 4 The Hydrogenlike Ion, Angular Momentum, and the Rigid RotorTABLE 4-2 Eigenfunctions for the Hydrogenlike Ion in Atomic Units
Spectroscopic symbol
Formula
√
1s (1/ π )Z3/2 exp(−Zr)
√
2s (1/4 2π )Z3/2(2 − Zr) exp(−Zr/2)
√
2px (1/4 2π )Z5/2r exp(−Zr/2) sin θ cos φ
√
2py (1/4 2π )Z5/2r exp(−Zr/2) sin θ sin φ
√
2pz (1/4 2π )Z5/2r exp(−Zr/2) cos θ
√
3s (1/81 3π )Z3/2(27 − 18Zr + 2Z2r2) exp(−Zr/3)
√
√
3px ( 2/81 π )Z5/2r(6 − Zr) exp(−Zr/3) sin θ cos φ
√
√
3py ( 2/81 π )Z5/2r(6 − Zr) exp(−Zr/3) sin θ sin φ
√
√
3pz ( 2/81 π )Z5/2r(6 − Zr) exp(−Zr/3) cos θ
√
3dz2 (≡ 3d3z2−r2 ) (1/81 6π )Z7/2r2 exp(−Zr/3)(3 cos2 θ − 1)
√
3dx2−y2 (1/81 2π )Z7/2r2 exp(−Zr/3) sin2 θ cos 2φ
√
3dxy (1/81 2π )Z7/2r2 exp(−Zr/3) sin2 θ sin 2φ
√
3dxz (1/81 2π )Z7/2r2 exp(−Zr/3) sin 2θ cos φ
√
3dyz (1/81 2π )Z7/2r2 exp(−Zr/3) sin 2θ sin φ Schr¨odinger equation in atomic units is (assuming infinite nuclear mass, so that µ = me)
−1∇2 − Z ψ = Eψ
(4-48)
2
r
The energies are En = − Z2
(4-49)
2n2
The lowest-energy solution is ψ1s = Z3/π exp(−Zr)
(4-50)
The formulas for the hydrogenlike ion solutions (in atomic units) of most interest in quantum chemistry are listed in Table 4-2. The tabulated functions are all in real, rather than complex, form. Problems involving atomic orbitals are generally far easier to solve in atomic units.
4-5 Angular Momentum and Spherical Harmonics
We have now discussed three problems in which a particle is free to move over the entire range of one or more coordinates with no change in potential. The first case was the free particle in one dimension. Here we found the eigenfunctions to be simple trigonometric or exponential functions of x. The trigonometric form is identical to the harmonic amplitude function of a standing wave in an infinitely long string. We might refer to such functions as “linear harmonics.” The second case was the particle-in-aring problem, which again has solutions that may be expressed either as sine-cosine or exponential functions of the angle φ. By analogy with linear motion, we could refer to these as “circular harmonics.” Finally, we have described the hydrogenlike ion, where the particle can move over the full ranges of θ and φ (i.e., over the surface of a sphere) with no change in potential. The solutions we have just described—the products l,m(θ)m(φ)—are called spherical harmonics and are commonly symbolized Yl,m(θ, φ). Thus for m 0
(2l + 1) (l − |m|)! 1/2 |
Y
m|
l,m(θ , φ) = (−1)m
P (cos θ ) exp (im φ)
(4-51)
4π
(l + |m|)!
l
and for m
of the set of associated Laguerre polynomials. Like the Legendre functions, these are mathematically well characterized. A few of the low-index associated Laguerre polynomials are L1(ρ) = 1, L1(ρ) = 2ρ − 4, 1
2
(4-47)
L1(ρ) = −3ρ2 + 18ρ − 18, L3(ρ) = −6
3
3
4-4 Atomic Units
It is convenient to define a system of units that is more natural for working with atoms and molecules. The commonly accepted system of atomic units for some important quantities is summarized in Table 4-1. [Note: the symbol ¯h (“h-cross or h-bar”) is often used in place of h/2π .] Additional data on values of physical quantities, units, and conversion factors can be found in Appendix 10.
In terms of these units, Schr¨odinger’s equation and its resulting eigenfunctions and eigenvalues for the hydrogenlike ion become much simpler to write down. Thus, the TABLE 4-1 Atomic Units
Values of some atomic properties in atomic units
Quantity
Atomic unit in cgs or other units (a.u.)
Mass me = 9.109534 × 10−28 g Mass of electron = 1 a.u.
Length
a0 = 4πε0 ¯h2/mee2
Most probable distance of 1s = 0.52917706 × 10−10 m electron from nucleus of H ( = 1 bohr) atom = 1 a.u
Time τ0 = a0 ¯h/e2 Time for 1s electron in H = 2.4189 × 10−17 s atom to travel one bohr = 1 a.u.
Charge e = 4.803242 × 10−10 esu Charge of electron = −1 a.u.
= 1.6021892 ×10−19 coulomb
Energy e2/4π ε0a0 = 4.359814 × 10−18 J
Total energy of 1s electron in ( = 27.21161 eV ≡ 1 hartree) H atom = −1/2 a.u.
Angular
¯h= h/2π
Angular momentum for momentum
= 1.0545887 × 10−34 J s particle in ring = 0, 1, 2, . . . a.u.
Electric field e/a2 = 5.1423 × 109 V/cm Electric field strength at
0
strength distance of 1 bohr from proton = 1 a.u.
Chapter 4 The Hydrogenlike Ion, Angular Momentum, and the Rigid RotorTABLE 4-2 Eigenfunctions for the Hydrogenlike Ion in Atomic Units
Spectroscopic symbol
Formula
√
1s (1/ π )Z3/2 exp(−Zr)
√
2s (1/4 2π )Z3/2(2 − Zr) exp(−Zr/2)
√
2px (1/4 2π )Z5/2r exp(−Zr/2) sin θ cos φ
√
2py (1/4 2π )Z5/2r exp(−Zr/2) sin θ sin φ
√
2pz (1/4 2π )Z5/2r exp(−Zr/2) cos θ
√
3s (1/81 3π )Z3/2(27 − 18Zr + 2Z2r2) exp(−Zr/3)
√
√
3px ( 2/81 π )Z5/2r(6 − Zr) exp(−Zr/3) sin θ cos φ
√
√
3py ( 2/81 π )Z5/2r(6 − Zr) exp(−Zr/3) sin θ sin φ
√
√
3pz ( 2/81 π )Z5/2r(6 − Zr) exp(−Zr/3) cos θ
√
3dz2 (≡ 3d3z2−r2 ) (1/81 6π )Z7/2r2 exp(−Zr/3)(3 cos2 θ − 1)
√
3dx2−y2 (1/81 2π )Z7/2r2 exp(−Zr/3) sin2 θ cos 2φ
√
3dxy (1/81 2π )Z7/2r2 exp(−Zr/3) sin2 θ sin 2φ
√
3dxz (1/81 2π )Z7/2r2 exp(−Zr/3) sin 2θ cos φ
√
3dyz (1/81 2π )Z7/2r2 exp(−Zr/3) sin 2θ sin φ Schr¨odinger equation in atomic units is (assuming infinite nuclear mass, so that µ = me)
−1∇2 − Z ψ = Eψ
(4-48)
2
r
The energies are En = − Z2
(4-49)
2n2
The lowest-energy solution is ψ1s = Z3/π exp(−Zr)
(4-50)
The formulas for the hydrogenlike ion solutions (in atomic units) of most interest in quantum chemistry are listed in Table 4-2. The tabulated functions are all in real, rather than complex, form. Problems involving atomic orbitals are generally far easier to solve in atomic units.
4-5 Angular Momentum and Spherical Harmonics
We have now discussed three problems in which a particle is free to move over the entire range of one or more coordinates with no change in potential. The first case was the free particle in one dimension. Here we found the eigenfunctions to be simple trigonometric or exponential functions of x. The trigonometric form is identical to the harmonic amplitude function of a standing wave in an infinitely long string. We might refer to such functions as “linear harmonics.” The second case was the particle-in-aring problem, which again has solutions that may be expressed either as sine-cosine or exponential functions of the angle φ. By analogy with linear motion, we could refer to these as “circular harmonics.” Finally, we have described the hydrogenlike ion, where the particle can move over the full ranges of θ and φ (i.e., over the surface of a sphere) with no change in potential. The solutions we have just described—the products l,m(θ)m(φ)—are called spherical harmonics and are commonly symbolized Yl,m(θ, φ). Thus for m 0
(2l + 1) (l − |m|)! 1/2 |
Y
m|
l,m(θ , φ) = (−1)m
P (cos θ ) exp (im φ)
(4-51)
4π
(l + |m|)!
l
and for m