Electronic Spectroscopy

An electronic transition involves absorption of a photon to produce a change in the electronic state of an atom or molecule. The hamiltonians for the ground and excited states are not the same in this case. Experimentally we measure the absorbance of a sample of molecules at known concentration. The absorbance is related to the molar extinction coefficient by Beer's law, A(w) = e(w)Cd, where C is the sample concentration and d is the path length of the cell. The definition of the extinction coefficient in terms of the absorption cross section is:

where e(w) is in units of M^-1cm^-1 and s(w) is in units of Å²/molecule. The absorption cross section is derived using the Fermi Golden rule of first order time-dependent perturbation theory.

It is usual to neglect the term e^-hw/kT relative to 1. Note that the sum over populations p_i is important only for systems where various levels are thermally populated. The term p_i represents e^-ei/kT/q. We can examine this in the Figures below.

In this Figure there are three vibrational levels in the ground state that are represented as being populated. There are transitions from each of these ground state levels (red) to excited state levels (blue). Other transitions are possible besides those shown. The above Figure is intended only to illustrate the concept that multiple transitions are possible. For the same system at T = 0 K there will only be one state populated as shown in the Figure below.

In this case p₀ = 1 and all other p_i = 0 since no higher vibrational states are populated.

We can express s(w) in time-dependent form

where K is given by

We will consider explicit transitions from the initial to the final state in which the hamiltonian is not the same. Radiation interactions with the electronic transition dipole to allow these transitions between states. In addition to the question of overall dipole strength the vibrational states that are affected by the change in electronic configuration give rise to a line shape for the electronic absorption band. In order to separate the contributions of the electronic nuclear wave functions several approximations are made. We can assume that each total wave function can be separated into an electronic and nuclear part

|iñ = |f_i(q;Q)ñ|c_i(Q)ñ

Here we have defined the electronic coordinates as q and the nuclear coordinates as Q. This approximation is known as the Born-Oppenhemier approximation. It states that electronic motion is rapid compared to nuclear motion and therefore the motions can be considered separately. A further assumption known as the Condon approximation assumes that the electronic wavefunction at the average nuclear coordinate is valid at all Q.

|iñ = |f_i(q;Q₀)ñ|c_i(Q)ñ

The Condon approximation treats the electronic wavefunction as a constant in the field of nuclear motion. It is also important in defining the properties of the electronic transition. By making the Condon approximation we assume that an electronic transition occurs very rapidly compared to nuclear motion and therefore that the coordinate Q is unchanged by the transition. This allows us to make a separation of electronic and nuclear wavefunctions. In the sum-over-states model this becomes

In the time-dependent method this becomes

The prefactor now has an additional term due to the square of the (orientation dependent) interaction of the transition dipole with the unit vector in the electric field direction. Note the definition of the transition dipole moment*** M_if = eáf_i|q|f_fñ as distinct from the permanent dipole moment in either the ground m_g = eáf_i|q|f_iñ or excited m_e = eáf_f|q|f_fñ states. The summation over initial and final states now applies only to the nuclear states. There is one initial electronic state and one final electronic state in this picture. The time-dependence is left in the nuclear part. Again, this stems from the Condon approximation which assumes that the electronic transition is rapid and nuclear motion follows. In the time-correlator method we must find the operator that transforms from the ground to the excited state vibrational manifold. This operator will propagate the ground state vibrational wavefunction using the excited state hamiltonian.

Line broadening

The Fermi Golden Rule is derived using a delta function implying that the linewidths of the individual electronic-vibrational transitions are infinitely narrow. It is possible to simply add width to the lines by applying a line shape function such as a Lorentzian, or Gaussian to represent the finite lifetime and dephasing time in the excited state. This has already been done implicitly for a Lorentzian in the definition of the sum-over-states cross section above. We define homogeneous broadening mechanisms as all those that act to broaden the spectrum of a single molecule. Inhomogeneous mechanisms are those that result from ensemble effects. A Lorentzian line shape function is derived for homogeneously broadened systems (the spectral line width is broadened by lifetime or pure dephasing processes). A Gaussian line shape function is derived for inhomogeneous broadening (the spectral line width is broadened due to site differences in condensed phase or Doppler shifts in the gas phase).

Methods for calculating the Franck-Condon lineshape

The sum-over-states method relies on individual calculation of the many possible overlap terms between the ground and shifted (Franck-Condon active) excited state. The disadvantage of the sum-over-states method is that each term in the series must be calculated explicitly and further the line shape broadening function is typically limited to Lorentzian. A more general form for the line shape function is possible using time-correlator methods. In the following discussion we consider linear displacements in the excited state with no frequency shifts.

Sum-over-states method

In the sum-over-states model given above we calculate the nuclear overlaps between the ground and excited state potential energy surfaces explicitly. Writing this out explicitly for a Lorentzian model for the dephasing we have

The evaluation of the overlap integrals has been derived by Manneback (Physica, 1951, 17, 1001).

where m and n are the vibrational levels of the excited and ground electronic states, respectively. Using the recursion relations the Franck-Condon factors FC = |ác_f|c_iñ|² = |ám|nñ|² can be calculated.

Time-correlator methods

In order to consider further time-evolution inherent in the nuclear factor we need a description for the nuclear hamiltonian in states i and f.

We define the initial state hamiltonian as

Where a+ and a are the creation and annihilation operators in the ground state manifold, respectively. The final state hamiltonian is defined as

We are interested in transitions between the two vibrational subsystems and therefore we seek the operator which effects the transformation from the initial state eigensystem a to the final state eigensystem b. Following the standard definition we can define the initial state operators as

where the position and momentum of the nuclei are represented by letters x and p, respectively. To represent the shift in nuclear coordinate that occurs we let the coordinate x shift by an amount z. Thus, the annihilation and creation operators become

We define the quantity

where D represents the displacement between the ground and excited state surfaces in units of the root-mean-square zero-point-displacement. Be comparison of the operators a and b in the ground and excited state manifolds, respectively, we see that b = a - D/Ö2. For convenience we will use d = D/Ö2 in the following derivations. We can show that the operator F = exp{(a⁺ - a)d} is the appropriate operator such that

FaF⁺ = b

We use the identity

e^ABe^-A = B + [A,B] + 1/2[[A,[A,B]] + …

(1 + A + 1/2A² + …)B(1 - A + 1/2A² + …) = B + AB - BA + 1/2A²B + 1/2 BA² + … giving the above commutators.

Thus, truncating after the first commutator,

exp{(a⁺ - a)d}a exp{(a - a⁺)d} = a + [(a⁺ - a)d,a]

= a + (a⁺ - a)da - a(a⁺ - a)d = a + (a⁺a - aa - aa⁺ + aa)d

= a + (a⁺a - aa⁺)d = a - d = b.

For the last step we used the property [a,a⁺] = 1.

We can express F in time-dependent form using the time-dependent definitions of the operators a_t = ae^-iwt and a_t⁺= a⁺eⁱwt.

Based on the definitions above we define the accepting mode correlation function as

C_a(t) = áF(0)F⁺(t)ñ

We refer to accepting modes as Franck-Condon active modes. These are vibrational modes that have shifts in nuclear coordinate associated with the electronic transition. The significance of the angle brackets is an expectation value or average in the ground state vibrational manifold. The reason for this is that we can use the definitions of F to write

The expectation value of C_a(t) can be computed by the use of the disentangling theorem and a cumulant expansion. The disentangling theorem states that for two operators A and B,

e^Ae^B = exp{A + B + 1/2[A,B]}

This allows the terms in áF(0)F⁺(t)ñ to be written as a single exponential.

exp{(a₀⁺ - a₀)d}exp{(a_t - a_t⁺)d} =

exp{(a₀⁺ - a₀)d - (a_t⁺ - a_t⁺)d + 1/2[(a₀⁺ - a₀)d ,- (a_t⁺ - a_t⁺)d]}

The commutator evaluates to

-d²/2(a₀a_t⁺ - a₀⁺a_t+ a₀⁺a_t⁺- a_ta₀+ a_ta₀⁺+ a_t⁺a₀- a_t⁺a₀⁺)

In order to evaluate the expectation value of áe^Cñ we use a cumulant expansion.

áe^Cñ = exp{ ááCññ + 1/2ááC²ññ + …}

where

ááCññ = áCñ

ááC²ññ = áC²ñ - áCñ²

Only terms in even powers of d will have non-zero expectation values. Thus, the terms in the commutator above will survive in áCñ. The linear terms will be squared in áC²ñ.

áCñ = á-d²/2(a₀a_t⁺ - a₀⁺a_t+ a₀⁺a_t⁺- a_ta₀+ a_ta₀⁺+ a_t⁺a₀- a_t⁺a₀⁺)ñ

áC²ñ = ád²(a₀²+ a_t⁺² - a₀a₀⁺- a₀⁺a₀- a₀a_t+ a₀⁺a_t+ a₀a_t⁺ - a₀⁺a_t⁺- a_ta₀+ a_t⁺a₀ + a_ta₀⁺- a_t⁺a₀⁺ + a_t²+ a_t⁺²- a_ta_t⁺- a_t⁺a_t)ñ

Once written in this form standard relations can be used

áa²ñ = 0 , áa⁺²ñ = 0 , áañ = 0 , áa⁺ñ = 0 , áa⁺añ = n , áaa⁺ñ = n + 1

where n = [exp{hw/kT} - 1]^-1.

Using these relations the expectation values of áCñ and áC²ñ are

áCñ = d²/2(ne^-iwt + (n+1)eⁱwt - neⁱwt - (n+1)e^-iwt)

áC²ñ = d²/2 (-2(2n + 1) + ne^-iwt + (n+1)eⁱwt + neⁱwt + (n+1)e^-iwt)

Collecting terms we obtain the result

C_a(t) = exp{ S[-(2n+1) + ne^-iwt + (n+1)eⁱwt] }

where S = d² = D²/2. The factor S is known as the linear electron-phonon coupling. The expression given is the correlation function required for the calculation of an electronic transition coupled to a single vibrational mode.

The expression for the transition probability can be written as a time-correlation function. This takes the form of the Fourier transform of the correlation function we have just derived.

There are several methods of solution, however, the numerical Fourier transform is the most preferable. Note that if there is more than one Franck-Condon active mode then the correlation function C_a(t) is a product of single-mode functions:

The wavepacket picture

The time-correlator formalism has been expressed various ways in the literature. In the wave-packet picture the time-correlator represents the loss of overlap of the initial ground state wave function with itself as it propagates on the excited state potential energy surface. This is illustrated below in one dimension.

The ground state is ái| and the time-dependent excited state is |i(t)ñ

so that the time-correlator becomes:

See Kulander and Heller (JCP, 1978, 69, 2439) and Heller et al. (JPC, 1982, 86, 1822). In this expression the dephasing is assumed to be exponential. The resultant line shape will be Lorentzian since the form of the above expression is a Fourier transform. The use of an exponential form for the dephasing is not necessary and solvent dynamics can be included in the expression using the Kubo function or other line shape function. In the general case the dephasing is given by the function exp{-g(t)}. The exponential function above is given by g(t) = |G|t.

By comparison with the above derivation of the accepting mode correlation function C_a(t) we can see that it is equivalent to ái|i(t)ñ. The description of the derivation of ái|i(t)ñ in McHale's "Molecular Spectroscopy" is formally equivalent to the detailed derivation here without the details that lead to a specific form that we have outlined here.

These methods are called the wavepacket picture. Imagine projecting the ground state wavefunction onto the excited state potential energy surface. Initially the overlap with the ground state wavefunction is unity. However, as the wavepacket moves (under the influence of the excited state hamiltonian) the overlap begins to decrease. The wavepacket begins oscillating and thus the overlap oscillates. Due to the exponential damping G the amplitude of the overlap decreases with each oscillation.

Calculations

Using the program TIMETHERM written by Shreve we can model absorption spectrum using the time-dependent formalism. We specify the following parameters for a single mode calculation:

G(homogeneous) = 50 cm^-1

G(inhomogeneous) = 200 cm^-1

w = 1200 cm^-1

w_eg = 20,000 cm^-1

T = 300 K

First the calculation is done for four values of the electron-phonon coupling constant S. As shown below ái|i(t)ñ has a generally exponential decay but has structure.

The structure can be envisioned as due to the wave function oscillating back and forth in the excited state and losing overlap or coming into overlap with the ground state wave function.

Imagine the wavepacket e^iHt/h|i(t)ñ moving out of overlap with ái|. This is the first rapid decay. Then the wavepacket reaches the other side of the excited state harmonic potential energy surface. This sends the wavepacket back until it reaches overlap again. However, during this time some amount of electronic state dephasing has taken place. This is determined by G as we discuss further below.

The Fourier transform of the correlation function gives the absorption spectrum as shown below.

Note that this calculation is indicated for four different values of the electron-phonon coupling constant. These values can be seen in the vibronic structure of the bands. For example, for S = 1 the 0-0 (20,000 cm^-1) and 0-1 (21,200 cm^-1) transitions are nearly equal in intensity.

Since the excited state dephasing width G = 50 cm^-1 for this calculation this corresponds roughly to a time constant of 200 femtoseconds for the decay. This estimate comes from the Uncertainty principle. As a rule of thumb Gt = 10 cm^-1ps. Thus, a G = 100 cm^-1 corresponds to a decay time of t = 100 femtoseconds etc. To illustrate the dependence of the correlation function ái|i(t)ñ on G we perform four representative calculations as a function of G (homogeneous).

Note that as G increases the decay rate increases and there are fewer oscillations. The Fourier transform of these correlation functions leads to the following spectra.

At a sufficiently high value of G (homogeneous) the absorption spectrum loses all structure.

Footnote

*** This function is given by eM_ifin Myers and Mathies treatment of absorption and Raman (Myers and Mathies in Biological Applications of Raman Spectroscopy, Vol. 2 page 16). We include the e (charge of the electron) in the definition of the transition moment M_if.