Molecular orbitals, potential energy surfaces and symmetryww2.sljus.lu.se/fyst20/lectures/Lecture 5 molecules.pdf · Molecular orbitals, potential energy surfaces and symmetry •

Molecular orbitals, potential energy surfaces and symmetry

• mathematical presentation of molecular symmetry– group theory– spectroscopy– valence theory– molecular orbitals

– Wave functions– Hamiltonian: electronic, nuclear wave functions

– Energy scales: 10-10000 eV, 0.1 eV, 0.01 eVH = Hel + Hvib + Hrot

Tuesday, 19 February 13

Molecular orbitals: constructed from atomic orbitals

• Atomic orbitals have spherical symmetry– basis set is spherical harmonics Ylm

• Molecular orbitals have symmetry determined by molecular geometry

– basis set is atomic orbitals, ϕA ϕB

– Pauli principle applies

• Effective bonding– energies of atomic orbitals are similar– overlap between orbitals strong– same symmetry with respect to rotation about

molecular bond axis


Molecular bonds: bonding and antibonding

Bonding orbital nonzeroelectron density between nucleiCONSTRUCTIVE INTERFERENCE

Antibonding orbitalNode (zero density) between nucleiDESTRUCTIVE INTERFERENCE

+


Energy level diagrams for homonuclear diatomic molecules

Li2 to N2O2 and F2


Energy level diagram for heteronuclear molecules

• Atomic orbitals ϕA ϕB

• s-orbitals• different atomic energy• form σ molecular orbitals•


Example: hydrogen molecular ion

• Calculate orbitals for the hydrogen molecular ion, H2+• Calculate potential energies for bonding and antibonding

sigma states• Use atomic 1s orbitals as a basis set• Effective charge described by ζ (= 1 for hydrogenic case)

The LCAO wave function in it’s simplest form


Basic strategy and equations

• Solve the SE using the Hamiltonian for a 3-body system (H1, H2, e-)

• Two wave functions, two equations• use symmetry to simplify• solve for energy eigenvalues• use atomic units...


Symmetry...

For a homonuclear molecule the basis wave functions are identical, and the coefficients in the LCAO wave function are equal. H11=H22

Two solutions: c1 =


Symmetry..

i gerade

i ungerade


Calculation of wave functions, potentials and other quantities...

• A brief MATLAB interlude...


What does it mean?


Polyatomics: Symmetry operations and elements

• I, identity• Cn, n-fold rotation

• σ Reflection (in a plane)

• i, inversion (through a center of symmetry)

• Sn, improper rotation (n-fold rotation in a plane followed by reflection through a plane perpendicular to rot plane)


Inversion

Rotation Reflection

Improper rotation

S = Cn . σ"


Symmetry groups

‘Lowest symmetry’ C1, Cs, CiCn: I, n-fold rotation (about principal axis)Cnv: I, n-fold rotation, vertical reflectionsCnh: I, n-fold rotation, horizontal reflectionDn: I, n-fold rotation, about principal and perpendicular

axisDnh: Dn and horiz reflection (homonuclear)Sn: I, n-fold improper rotation

C2v


Calculus of symmetry elements

• Symmetry elements generate other elements: C32 = C3 x C3

All symmetry elements are generated from others

Matrix representationGroup theory applies to molecular

symmetries•


Character tables

• Symmetry classification of properties of molecules (wave functions, orbitals, states)

DegeneracySymmetry elementsBasis for constructing orbitals, vibrational

motion, electronic transitions


Example: C2v point group character table (water, SO2, etc)

I C2 σv σv� A1 1 1 1 1 A2 1 1 -1 -1 B1 1 -1 1 -1 B2 1 -1 1 -1

Irreducible representation �of symmetry species�

Generating elements�


Heteronuclear diatomics

I C∞ ... σv A1 1 1 … 1 A2 1 1 … -1 E1 2 2cosφ … 0 E2 2 2cos2

φ$... 0

Σ+ Σ� Π Δ


Molecular bonds: σ symmetry


Molecular bonds: π symmetry


Potential energy function

Neutral oxygen molecule


Symmetry adapted Linear Combination AO

• Example: water C2v symmetry

I C2 σv(xz) σv(yz)A1 1 1 1 1

A2 1 1 -1 -1

B1 1 -1 1 -1

B2 1 -1 -1 1


1/2（1sA + 1sB）

Construct wave functions-following the rules

I C2 σv(xz) σv(yz)O 2s 1 1 1 1O 2px 1 -1 -1 1O 2py 1 -1 1 -1O 2pz 1 1 1 1H 1sA 1 H 1sB 1 H 1sB

H 1sB 1 H 1sA 1 H 1sA

BA

1/2（1sA - 1sB）

B2

A1

A1

A1

B1

B2


Ψ(A1) = c1 O 2s + c2 O 2pz + c3 ϕ (A1)

Symmetry adapted wave functions

Ψ(B1) = O 2py

Ψ(B2) = c4 O 2px + c5 ϕ (B2)

where the symmetry adapted molecular orbitals are

ϕ (A1) = 1/2（1sA + 1sB）ϕ (B2) = 1/2（1sA - 1sB）


Molecular orbitals for water


Molecular orbital labels

• Water molecule: 8 oxygen electrons 2 hydrogen 1s

B2 ... b2

A1 ... a1

B1 ....b1

O1s21a12 1b22 2a12 1b12 1A1

Ground-state configuration and term

1b1

1b2

3a1

2a1

Bond angle 2`

90k 120k 150k 180k

bent linear

4a1

2mg

1mu

1/u

3mg

XH2 molecules

Walsh diagram


Vibrational modes

A1 B1

Symmetric antisymmetric


Quantum mechanical treatment of vibrations in molecules

The full wave function is a product of electronic and nuclear wave functions (Born-Oppenheimer approx)

electrons (r) immediately ‘follow’ the nuclear motion (R)

Break down matrix element to electronic and nuclear transition elements


Quantum mechanical treatment of vibrations in molecules• Attractive restoring force (approximate harmonic osc)• .... solve the Schrödinger equation:

With eigenfunctions based upon Hermite polynomials

and eigenfrequencies:

Table 2.1: Hermite polynomials.v H

v

(y) v Hv

(y)0 1 3 8y3 � 12y1 2y 4 16y4 � 48y2 + 122 4y2 � 2 5 32y5 � 160y3 + 120y

can have even at absolute temperature of 0 K. It is a result of the uncertaintyprinciple.

The crossing points of an energy level with the potential energy curvecorrespond to the classical turning points of vibration, where the velocitiesof the nuclei are zero and all energy is in the form of potential energy. In themiddle point of each energy level all energy is conversely kinetic energy.

The solutions of Eq. (2.4) are wavefunctions

v

=✓

1

2vv!⇡1/2

◆1/2

Hv

(y) exp(�y2/2), (2.7)

where Hv

(y) is a Hermite polynomials and

y =

4⇡2⌫µ

h

!1/2

(r � re

). (2.8)

Some of the Hermite polynomials are given in Table 4.1. Examples ofvibrational wavefunctions are presented in Fig. 4.1. We note the followingimportant properties of the wavefunctions:

1. They extend to the region outside of the parabola, which is forbiddenin a classical system.

2. When v increases those two points where the probability density 2

reaches its maximum value occur close to the classical turning points.This is illustrated in Fig. 4.1 for the quantum number v = 28, withA and B being the classical turning points. In contrast, for v = 0 thehighest probability density is in the middle of the region.

The force constant k can be regarded as a measure for the strength of thebond. Table 4.2 gives some typical values in the units of aJ A�2(= 102 N/m).The values describe how k increases with the bond order. Molecules HCl,

19


v

(y) v Hv

(y)0 1 3 8y3 � 12y1 2y 4 16y4 � 48y2 + 122 4y2 � 2 5 32y5 � 160y3 + 120y




v

=✓

1

2vv!⇡1/2

◆1/2

Hv

(y) exp(�y2/2), (2.7)

where Hv


y =

4⇡2⌫µ

h

!1/2

(r � re

). (2.8)






19

Figure 2.1: The energy levels, wavefunctions and potential energy V (r) ofthe harmonic oscillator.

where µ = m1m2/(m1+m1) is the reduced mass of the atoms. The Schrodingerequation for the system is

d2 v

dx2+

2µE

v

h2 � µkx2

h2

!

v

= 0. (2.4)

It can be shown thatE

v

= h⌫(v + 1/2), (2.5)

where ⌫ is the classical frequency of the oscillator, which can be obtainedfrom

⌫ =1

2⇡

k

µ

!1/2

. (2.6)

As expected, the frequency increases when the bond becomes stronger (whenk increases) and decreases when µ becomes larger. The vibrational quantumnumber v can have values 0, 1, 2, ... .

According to Eq. (2.5) the vibrational energy levels are spaced by a con-stant interval of h⌫ and the lowest energy of the oscillator (when v = 0) isnot zero but 1

2h⌫. This zero-point energy is the lowest energy that a molecule

18


v

(y) v Hv

(y)0 1 3 8y3 � 12y1 2y 4 16y4 � 48y2 + 122 4y2 � 2 5 32y5 � 160y3 + 120y




v

=✓

1

2vv!⇡1/2

◆1/2

Hv

(y) exp(�y2/2), (2.7)

where Hv


y =

4⇡2⌫µ

h

!1/2

(r � re

). (2.8)






19


Vibrational progressions

€

Eν = E0 + ω0(ν +1/ 2)+ ω0xe(ν +1/ 2)2

Adiabatic binding energy

Harmonic term Anharmonic term


Molecular potential function, vibrations and dissociation

Figure 2.2: The potential energy curve and energy levels of a diatomicmolecule when it behaves as an anharmonic oscillator. Dashed curves givethe same properties for a harmonic oscillator.

longer. Then the force constant is zero and r can be increased to infinitywithout influencing the potential energy V . This is illustrated in Fig. 4.2.The potential energy curve levels at a value V = D

e

, where De

is the dis-sociation energy as measured from the potential energy at the equilibriumdistance. Thus for r > r

e

, the potential energy becomes lower than in thecase of the harmonic oscillator. At small values of r, the positive chargesof the nuclei cause a repulsion that opposes bringing the nuclei closer eachother and the potential energy curve is steeper than that of the harmonicoscillator.

Anharmonicity changes the wavefunctions and term values. The termvalues of the harmonic oscillator in Eq. (4.9) become a power series in (v +1/2)

G(v) = !e

(v +1

2)� !

e

xe

(v +1

2)2 + !

e

ye

(v +1

2)3 + . . . (2.10)

where !e

is the vibrational wavenumber that a classical oscillator wouldhave infinitely close to the equilibrium position. !

e

xe

, !e

ye

. . . are anhar-monicity constants. The terms !, !

e

xe

, !e

ye

. . . in the series (2.10) become

21


Franck-Condon principle• Absorption of radiation-vibronic

transitions• Electronic transition from state with

energy E’’el and vibrational quantum number v’’=0 to an excited state E’el

• The electronic state transition is independent of the vibrational transition

• Electronic state transition: DIPOLE selection rules

• Vibronic transition, no strict selection rules

• E’’el(v’’) --> E’el(v’)Figure 3.4: The Franck-Condon principle applied to the case r0

e

> r00e

whenthe 4� 0 transition is the most probable.

33


Frank-Condon principle: intuitive picture

Figure 3.4: The Franck-Condon principle applied to the case r0e

> r00e

whenthe 4� 0 transition is the most probable.

33

Franck-Condon region is defined by the ground-state wave function

Absorption from the ground state is strongest to the wave function in the excited state that has a maximum in the center of the Franck-Condon region


Franck-Condon principle: quantum mechanical treatment

€

S = Ψ0∫ (x)Ψ'o (x)dx

€

S0−0 = e−α4

2

Re − Re'( )2

€

Δq = 2 ⋅ S ⋅ µωgr

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2ωgr

ω fin

Transition moment for vibrational transitions

general form of vibrationalwave functions

Hv are Hermite polynomials, and α = 2π/h √mhv


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Molecular orbitals, potential energy surfaces and symmetryww2.sljus.lu.se/fyst20/lectures/Lecture 5 molecules.pdf · Molecular orbitals, potential energy surfaces and symmetry •