Appendix#

Point group flowchart#

../_images/flowchart.png

Fig. 1 Flowchart for the determination of molecular point groups.#

Character tables#

C2h#

Table 1 Character table for the C2h point group.#

Irrep

E^

C^2(z)

i^

σ^h

Operation

Ag

1

1

1

1

Rz, x2, y2, z2

Bg

1

-1

1

-1

Rx, Ry

Au

1

1

-1

-1

z

Bu

1

-1

-1

1

x, y

D2h#

Table 2 Character table for the D2h point group.#

Irrep

E^

C^2(z)

C^2(y)

C^2(x)

i^

σ^(xy)

σ^(xz)

σ^(yz)

Operation

Ag

1

1

1

1

1

1

1

1

x2, y2, z2

B1g

1

1

-1

-1

1

1

-1

-1

Rz, xy

B2g

1

-1

1

-1

1

-1

1

-1

Ry, xz

B3g

1

-1

-1

1

1

-1

-1

1

Rx, yz

Au

1

1

1

1

-1

-1

-1

-1

B1u

1

1

-1

-1

-1

-1

1

1

z

B2u

1

-1

1

-1

-1

1

-1

1

y

B3u

1

-1

-1

1

-1

1

1

-1

x

Atomic units#

A central problem in molecular physics is to solve the time-independent Schrödinger equation for the electrons in the field of the nuclei. Most often atomic units are then adopted. For the hydrogen atom, we have

[22me2e24πε0r]ψ(r)=Eψ(r)

where is the reduced Planck constant, me is the electron mass, e is the elementary charge, and ε0 is the electric constant. To cast this equation into dimensionless form, consider a coordinate transformation of the form

r=(x,y,z)λr=(λx,λy,λz)

to arrive at

[22meλ22e24πε0λr]ψ(r)=Eψ(r)

Choose λ so that

2meλ2=e24πε0λEh

with the solution

λ=24πε0mee2a0;Eh=mee4(4πε0)22

With E=E/Eh, we get

[1221r]ψ(r)=Eψ(r)

with a solution for the ground state energy that is equal to E=0.5 a.u. (or Hartree). The defined quantity a0 is equal to the Bohr radius and the atomic unit of length is therefore also referred to as Bohr.

Table: Atomic unit conversion factors.

Quantity

Symbol

Atomic unit

SI equivalent

Energy

E

1 Eh

4.359 744×1018 J

Reduced Planck constant

h=2π

1

1.054 572×1034 J s

Time

t

1 Eh1

2.418 884×1017 s

Length

l

1 a0

5.291 772×1011 m

Speed of light

c

137.036 a0Eh1

2.997 925×108 m s1

Electric constant

ε0

1 4πε0

8.854 188×1012 F m1

Fine structure constant

α

1/137.036 e2(a0Eh4πε0)1

7.297 353×103

Charge

q

1 e

1.602 176×1019 C

Electric field

F

1 Eh(ea0)1

5.142 207×1011 V m1

Dipole moment

μ

1 ea0

8.478 353×1030 C m

Mass

m

1 me

9.109 383×1031 kg