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Electronic Spectroscopy of diatomic molecules

Born - Oppenheimer approximation: It is supposed that the nuclei, being so much heavier than an electron, move relatevely slowly and may be treated as stationary while the electrons move in their field. We can therefore of the nuclei as being fixed at arbitrary locations and then solve the Schr¨dinger equation for the wave function of o the electrons alone. Molecular bond theory (MO): Electron should not be regarded to particular bonds but should be treated as spreading throughout the entire molecule. The Hamiltonian for H+ is: 2 e2 1 1 1 H=- (1) - ( + - ) 2me 4 0 rA1 rB1 R The Schr¨dinger equation can be solved only for hydrogen ion, o therefore the approximation of linear combination of atomic orbitals (LKAO) is used.

2 1 2

MO wavefunction

± = N (A ± B) For + wawefunction

2 + = N 2(A2 + B 2 + 2AB)

(2) (3)

It is bonding orbital

Figure 1: The amplitude of the bonding molecular orbital in a hydrgen molecule ion in a plane containing to nuclei and a contour trepresentation of the amplitude.

Unbonding orbital

The overlap density is critical, because it represent an encancement of the probability of finding the electron in the internuclear region. The echancement can be traced to the constructive interference of the two atomic orbitals. For unbonding orbital this interference is destructive.

Figure 2: The amplitude of the antibonding molecular orbital in a hydrgen molecule ion in a plane containing to nuclei and a contour trepresentation of the amplitude.

Figure 3: The electron density of the unbonding orbital

Figure 4: The calculated and experimental molecular potential energy curves for a hydrogen molecule ion

Figure 5: Partial explanation of the origin of bonding and unbonding effects

Figure 6: A molecular orbital diagram for hydrogen molecule

Figure 7: A molecula orbital diagram for helium

Bond order

1 (4) b = (n - n) 2 where n is number electrons in bonding orbitals and n* is the nuber of electrons in atibonding orbitals.

Bond length and bond energies

Bond HH NN HCl CH CC CC CC CO FF

Order 1 3 1 1 1 2 3 2 1

re/° A 0.74 1.09 1.27 1.14 1.54 1.34 1.2 1.07 1.42

D0/kcal mole-1 103.3 225.0 102.2 104 88 172 230 256.7 37

pz -orbitals can produce molecular orbitals

Figure 8: A representation of the composition of bonding and antibonding orbitals built from the overlap of p orbitals

pz -orbitals can produce orbitals

Figure 9: A scematic representation of the structure of bonding and antibonding orbitals

Figure 10: The molecular orbital diagram for homonuclear diatomic molecules of period 2 up to and including N2

Figure 11: The molecular orbital diagram for homonuclear diatomic molecules for O2 and F2

Energy diagram for homonuclear molecules of the second period

Parity for homnuclar molecules

Figure 12: The parity of an orbital is even (g) if its wavefunction is unchanged under inversion through the centre of symmetry of molexule and odd (u) if the wavefunction changes sign.

Symmetry according to reflection

Figure 13: The + or - on a term symbol refers to the overall symmetry of a configuration under reflection in a plane containing the two nuclei. Used only for terms.

Hund's case (a)

Figure 14: Spin orbit interaction is very weak.

Term symbols

· The value of (the component of orbital angular momentum L about the internuclear axis) is denoted by symbols , , , etc for || = 0, 1, 2 ... · As in atoms a superscrift with the value of 2S+1 is used to denote the multiplicity of term · For homonuclear molecules the overall parity is added as a right subscript. If there are several electrons the overall parity is calculated by using g×g =g u×u=g u×g =u (5)

· For > 0 the value of || + is also noted as a right subscript

The electronic states of oxygen

Selection rules

· = 0, ±1

Selection rules

· = 0, ±1 · S = 0

Selection rules

· = 0, ±1 · S = 0 · = 0 (for Hund's case (a))

Selection rules

· = 0, ±1 · S = 0 · = 0 (for Hund's case (a)) · = 0, ±1

Selection rules

· = 0, ±1 · S = 0 · = 0 (for Hund's case (a)) · = 0, ±1 · + + or - - but + and - -. Note that +

Selection rules

· = 0, ±1 · S = 0 · = 0 (for Hund's case (a)) · = 0, ±1 · + + or - - but + and - ·gu -. Note that +

Labeling of electronic states

There is a convention for labeling of the electronic states. In diatomic molecule the groun state is labeled as X and the higher states of the same multiplisity are labeled as A, B, C.... States with multiplisity different from that of the ground state are labeled as a, b, c.... In poliatomic molucules the symmetry of states are noted by the latin letters and not by the greek leters as for diatomic ones. There~ b, ~ fore the states in this case are labeled as X, A, B, C..., a, ~ c...

Vibrational structure

An electronic transition is made up of vibrational bands, each ofwhich is in turn made up of rotational lines. The presense of many vibrational bands, labeled as v - v explains why electronic transition are often called band system. The vibration structure is organized into sequences and progressions. A group of bands with the same v is called a sequence so that the 0-0, 1-1 2-2 bands form v = 0 sequence. A series of bands all connected to the same vibrational level v, such as 3-1, 2-1, 1-1, 0-1 is called a progession.

Intensity of the vibrational bands

The intensities of the various vibrational bands are determined by three factors: · The inrinsic strength of the electronic transition · The poulation of the initial vibration level · The squared overlap integral of the two vibrational wavefunctions, called the Franck-Condon factor

Classical version of the Franck-Condon principle

Figure 15: The most intense vibronic transition is from the ground vibrational state to the vibrational state lying vertically above it. Transition to other vibrational levels also occur, but with lower intensity.

The quantum mechanical version of the Franck-Condon principle

The intesity of given transition is proportional to the square of the transition moment integral: M ev =

e v µe v

d

(6)

In the Born-Oppenheimer approximation ev = ev The rotational motion of the diatomic molecule is ignored here. The dipole moment operator can be broken into electronic part and nuclear part: µ= qiri =

j

qj rj +

k v v dN +

qk rk = µe + µN

e e del

(7)

M=

e µee del

v µN v dN

(8)

M=

e µee del +

e e del

v µN v dN

The last term in the equation is zero sinse the electronic wavefunctions of two different states are orthogonal. Finally M = Re < v |v > where < v |v" >= and Re =

e µee del v v dN

(9) (10) (11)

Thus the intensity of a vibronic transition is Ie v e where qv -v = | < v |v > |2 (13)

v

|Re|2qv -v

(12)

Quantum treatment of the Franck-Condon principle

Figure 16: In the quantum mechanical version of the Franck-Condon principle, the molecule undergoes a transition to the upper vibrational state that most closely resembles the vibrational wavefunction of the vibrational ground state of the lower electronic state. The two wavefunctions shown here have the gratest overlap integral and hence are most closely simmilar

Figure 17: The Franck-Condon factor for 0-0 transition and Morse potentials.

Rotational structure of electronic transitions of diatomic molecules

For electronic transition the rotational constants for P, Q and R branches can differ significantly.: P = 0 - (B + B )J + (B - B )J 2 R = 0 + 2B + (3B - B )J + (B - B )J 2 Q = 0 + (B - B )J(J" + 1) (14) (15) (16)

The energy expression for P and R branches can be combined into a single expression nuP,R = 0 + (B + B )m + (B - B )m2 where m=J"+1 for the R branch and m=-J" for the P branch (17)

Formation of the bandhead

Formation of the bandhead

Formation of the bandhead

Formation of the bandhead

Formation of the bandhead

Formation of the bandhead

Formation of the bandhead

Formation of the bandhead

Position of the bandhead

(B + B ) mH = - (18) 2(B - B ) (B + B )2 (19) H - 0 = - 4(B - B ) Thus, if B¡B then the band shaded (degraded) toward the red if B¿B then the band shaded (degraded) toward the violet If B B the head lies at such a great distance from the band origin that is not observed.

Figure 18: Typical vibrational progession intensity distribution

Deslandres Table

Figure 19: Deslandres table of the bandheads of the N2 C 3 u - B 3 g second positive system

Singlet-Singlet transitions, S=0

The rotational structure of singlet-singlet electronic transition is identical to that of the vibrational transition of linear molecule. · = 0, = = 0. These transition have only P and R branches (J = ±1). 1 -1 are paralel transitions with the transition dipole moment lying along the z-axis · = ±1 These transitions have strong Q branches as well as P and Q branches, with J = 0, ±1. These transitions are perpendicular transitions. · = 0, but = = 0 Transition such as 1 -1 or 1 -1 are characterised by weak Q branches and strong P and R branches

Figure 20: The 0-0 band of the CuD A1 + - X 1 + system. The peaks marked with + are P branch transitions, while those marked with * belong to the R branch

Figure 21: The 0-0 band of the NH c1 - a1 system. The doubling of the lines in the R and Q branches is due to doubling in the c1 state. The very intensive line near 30790 cm-1 is due emission from the He atom

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