10.9 Harmonic Vibrational Analysis

10.9.5 Vibrationally-Resolved Electronic Spectra and Resonance Raman Simulations

(May 16, 2021)

Optical spectroscopy is the study of the interaction between the light and matter, and the study that encompasses a wide range of physical and chemical behavior, which can be directly recorded by the spectrometers. Contemporary spectroscopic techniques have been applied to widespread research fields and have served as a popular tool to obtain the information of structural and dynamical features of the matter. However, the experimentally-measured spectra can’t straightforwardly give the microscopic information of the matter. The theoretical calculations of the spectra can serve as a supplementary tool to the experimental measurements and provide a deeper understanding on the underlying physical and chemical phenomena. 838 Pedone Alfonso, Biczysko Malgorzata, Barone Vincenzo
ChemPhysChem
(2010), 11, pp. 1812.
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, 63 Barone Vincenzo et al.
Phys. Chem. Chem. Phys.
(2012), 14, pp. 12404.
Link
One can easily determine geometrical, electronic and dynamical features of matters through a comparison between the experimental results and the calculated values. Moreover, the role of different effects in spectroscopic properties can also be quantified by the calculations.

Vibrationally-resolved one-photon absorption (OPA) and emission (OPE) spectra and resonance Raman scattering (RRS) spectra, each of which involves simultaneous changes in the vibrational and electronic states of a molecule, can reveal a reliable molecular structure–property relationship. Theoretical prediction of these spectra needs to combine both the electronic structure theories and quantum dynamics methods to obtain the structure parameters and describe quantum dynamics, respectively. 1028 Spiro T. G., Stein P.
Annu. Rev. Phys. Chem.
(1977), 28, pp. 501.
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, 1152 Warshel A.
Annu. Rev. Biophys. Bioeng.
(1977), 6, pp. 273.
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, 190 Champion P. M., Albrecht A. C.
Annu. Rev. Phys. Chem.
(1982), 33, pp. 353.
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, 778 Myers A. B.
Chem. Rev.
(1996), 96, pp. 911.
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, 779 Myers A. B.
Acc. Chem. Res.
(1997), 30, pp. 519.
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, 1141 Wächtler M. et al.
Coord. Chem. Rev.
(2012), 256, pp. 1479.
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For RRS simulation using the IMDHO model (which neglects Duschinsky rotation), see Section 10.9.4.

10.9.5.1 Time-dependent approach to simulating spectra

On the basis of perturbation theory, the transition rate of one- or two-photon transition processes from the initial state |I to the final state |F is proportional to kIF=|F|M^|I|2δ(Δω), where δ(Δω) is the lineshape function with Δω=ωi-ωFI for a one-photon transition and Δω=ω1+ω2-ωFI for a two-photon process. Here ωi, ω1, and ω2 denote the incident photon frequencies and the operator M^ is given by

M^={μ^(one-photon transition)L[μe^2|LL|μe^1(ω1-ωLI)+μe^1|LL|μe^2(ω2-ωLI)](two-photon transition). (10.26)

In the two-photon case there are intermediate or “virtual” states |L.

Raman scattering is a two-photon process. In this process, one photon with the frequency ωi is absorbed, another photon with the frequency ωS is emitted, and the transition from the initial to the final vibrational states takes place. Based on perturbation theory, the transition rate of the Raman process is proportional to S(ωi,ωS)=|F|M^|I|2δ(Δω), where

M^=L[μe^2|LL|μe^1(ωi-ωLI)-μe^1|LL|μe^2(ωS+ωLI)] (10.27)

and Δω=ωS-ωi+ωFI. The differential photon scattering cross section is given by 34 Albrecht A. C.
J. Chem. Phys.
(1961), 34, pp. 1476.
Link

σ(ωi,ωS)4ωiωS39c4S(ωi,ωS). (10.28)

RRS spectroscopy is a type of vibrational Raman spectroscopy in which the incident laser frequency is close to an electronic transition of the molecule or crystal studied. As the adiabatic energy gap ωLI between the L state and the initial I state is close to the laser frequency ωi, the intermediate L state will make the dominant contribution to RRS. Under the “resonant” condition, the contributions from the non-resonant electronic states can be neglected.

One may evaluate MIF=ΦF|M^|ΦI by making use of the Herzberg-Teller (HT) expansion, i.e., one expands the integrals about the nuclear equilibrium configuration Q=0. Writing the pure-spin Born–Oppenheimer (psBO) functions as products of an electronic wavefunction Ψ and a vibrational wavefunction Λ

Φn(q,Q)=Ψn(q,Q)Λn(Q), (10.29)

we have

Ψn(q,Q)=Ψn(q,0)+[Ψn(q,0)/Q]Q=0Q+,MIF=MIF(Q=0)+[MIF/Q]Q=0Q+. (10.30)

The second term in Eq. (10.30) origins from the HT expansion. If we truncate the expansions after the lowest-order non-vanishing term, MIF can be written as

MIF =ΨF(q,0)|M^|ΨI(q,0)ΛF(Q)|ΛI(Q)
  +[(/Q)ΨF(q,0)|M^|ΨI(q,0)]Q=0ΛF(Q)|Q|ΛI(Q). (10.31)

If the first term, the direct transition, vanishes, this process is orbitally forbidden.

To evaluate the vibrational terms in the remaining part of the vibronic matrix elements, we can use the harmonic oscillator approximation. Then the vibrational part of the wave function is written as Λn=k=1Nχk(n)(νn), where N is the total number of normal modes χk and νn the vibrational quantum number associated with mode k in state |n.

The delta function δ(Δω) can be expressed as the Fourier integral

δ(Δω)=12π-+e(iΔω)t/𝑑t, (10.32)

and then the transition rate from the initial state to the final state becomes

kIF=-+exp[i(ω0+Ei-Ef)t/-γt]Ci(t)𝑑t (10.33)

where γ is a damping factor and

Ci(t)=tr[e-βH^ieiH^it/M^e-iHft/M^)tr[e-βH^i]. (10.34)

Here β=1/kBT, ω0=ωi in one-photon absorption and emission processes, ω0=ωi-ωS in Raman scattering process. The notation tr() represents a trace over nuclear and electronic degrees of freedom, and M^=|ΛiMIFΛf|+|ΛfMFIΛi|. The quantities H^i and H^f denote the nuclear Hamiltonians of electronic ground and excited states, respectively.

The Hamiltonian of vibrational motions on the ground and excited states can be written as

Hg =12jN[(Pg,i)2+(ωjgQg,j)2], (10.35a)
He =12jN[(Pe,i)2+(ωjeQe,j)2], (10.35b)

where P and Q are the momenta and coordinates of vibrational normal modes, respectively. The normal mode coordinates of ground and excited states are correlated by the Duschinsky rotation matrix D¯, with Qe=D¯Qg+Δ¯. The quantity Δ¯ is the displacement of normal mode coordinates between ground and excited states, i.e., the same quantity that appears in the IMDHO theory of Section 10.9.4, Eq. (10.20). The dimensionless forms are correspondingly Δj=(ωje)1/2Δ¯j and Dij=(ωie/ωjg)1/2D¯ij. From the above, the transition rate can be calculated directly in the time domain using the correlation function approach. This time-dependent approach has been implemented to calculate vibronic spectra. 703 Ma H., Liu J., Liang W.
J. Chem. Theory Comput.
(2012), 8, pp. 4474.
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, 669 Liang W. et al.
J. Phys. Chem. B
(2006), 110, pp. 9908.
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, 704 Ma,HuiLi, Zhao,Yi, Liang,WanZhen
J. Chem. Phys.
(2014), 140, pp. 094107.
Link
, 666 Liang W. et al.
Int. J. Quantum Chem.
(2015), 115, pp. 550.
Link

It is obvious that ground and excited electronic states have different potential energy surfaces (PES) which lead to different vibrational frequencies and normal modes. The relation between mass-weighted Cartesian displacement coordinates x and normal mode coordinates Q is given by

xg-x0g=LgQg,xe-x0e=LeQe, (10.36)

where x0g and x0e are the equilibrium structures of ground and excited states. For an ideal N-dimensional harmonic oscillator, the normal mode coordinates of ground and excited states are related by

Qe=(Le)TLgQg+(Le)T(x0g-x0e)=D¯Qg+Δ¯. (10.37)

The minimum points at the PES and the Hessian matrix are required to calculate the Duschinsky rotation matrix and displacement vector. It can be time-consuming to calculate the excited state PES, especially for large molecules. The linear coupling model (LCM), which is also known as the vertical gradient (VG) approximation, has been proposed to avoid this issue. 175 Cederbaum L. S., Domcke W.
J. Chem. Phys.
(1976), 64, pp. 603.
Link
Assuming that the excited state PES is approximated by a shift in the ground state PES, namely ωje=ωjg and Le=Lg, the displacement of Qj can be calculated by the excited state energy gradient (EQ)j, and ΔVG can be written as

ΔjVG=(ωjg)-3/2(EQ)j=i(ωjg)-3/2(Exi)Lgij. (10.38)

The VG approximation is equivalent to the IMDHO approximation that is discussed in Section 10.9.4. 531 Kane K. A., Jensen L.
J. Phys. Chem. C
(2010), 114, pp. 5540.
Link
, 393 Guthmuller J.
J. Chem. Phys.
(2016), 144, pp. 064106.
Link

Generally, the Franck-Condon (FC) approximation is accurate enough for strongly one- or two-photon allowed transitions, while it breaks down for forbidden or weakly allowed transitions, and the FC term becomes nearly zero. In this situation, a correction to this deviation should be introduced by including the Herzberg-Teller (HT) or non-Condon effect. 672 Lin S. H., Eyring H.
Proc. Natl. Acad. Sci. USA
(1974), 71, pp. 3802.
Link
, 821 Orlandi G, Siebrand W
J. Chem. Phys.
(1973), 58, pp. 4513.
Link
, 34 Albrecht A. C.
J. Chem. Phys.
(1961), 34, pp. 1476.
Link
, 33 Albrecht A. C., Hutley M. C.
J. Chem. Phys.
(1971), 55, pp. 4438.
Link
, 1018 Small Gerald J
J. Chem. Phys.
(1971), 54, pp. 3300.
Link
HT-type vibronic coupling comes from the normal mode-coordinate dependence of the transition moments. When these quantities are expanded in terms of the normal mode coordinates, the contribution of the linear-coordinate-dependent terms is commonly called HT effect. 34 Albrecht A. C.
J. Chem. Phys.
(1961), 34, pp. 1476.
Link
, 33 Albrecht A. C., Hutley M. C.
J. Chem. Phys.
(1971), 55, pp. 4438.
Link
Many works, whether or not they account for the mode-mixing or Duschinsky rotation (DR) effect or not, have shown the importance of the HT effect in OPA, OPE, and RRS spectra. 666 Liang W. et al.
Int. J. Quantum Chem.
(2015), 115, pp. 550.
Link

To predict OPA, OPE and RRS spectra, electronic structure calculations on ground and excited states should be performed. The necessary jobs at different level of approximation are summarized in the following:

  • FC. This is available for OPA, OPE, and RRS spectra.

    1. 1.

      x0gx0e, Lg=Le, and ωg=ωe. Geometry optimization on excited state PES is performed, followed by ground state optimization and frequency analysis.

    2. 2.

      x0gx0e, LgLe, and ωgωe. Geometry optimization and frequency calculation are needed on both ground and excited states.

  • FCHT. This is available for OPA, OPE, and RRS spectra.

    • x0gx0e, LgLe, and ωgωe. It is similar to the second kind of FC calculation, in which transition dipole derivative is obtained via frequency calculation on excited state. Geometry optimization and frequency calculation are needed on both ground and excited states.

  • VG. This is available for OPA and RRS spectra.

    • x0g=x0e, Lg=Le, and ωg=ωe. Only the geometry optimization and frequency calculation of the ground state is involved. Frequencies and normal modes of excited state are assumed to be the same as ground state. The displacement vector is approximated by Eq. (10.38), in which the gradient of excited state PES is produced by excited state force job. Of course VG model has only contribution from FC term.

10.9.5.2 Job Control

Since both ground state and excited state parameters are required, the routines to predict vibronic spectra are designed to have two steps. Firstly excited state calculation is performed and information about excited state will be saved in $QCSCRATCH/savename. Then the vibronic spectra utility is called to simulate the requested spectra after frequency analysis on ground state. SYM_IGNORE should be set to TRUE in order to prevent the molecular geometry being transformed to the standard orientation. Therefore the vibronic spectra job input can be set up in two ways. First, the multiple jobs can be separated by the string @@@ as described in Section 3.5. Or, jobs can be separated into individual inputs using $QCSCRATCH/savename as described below and given as examples 10.9.5.3 and 10.9.5.3.

qchem infile_excited_state outfile_excited_state savename
qchem infile_ground_state outfile_ground_state savename

There are two $rem variables and one section $vibronic involved in vibronic spectra calculations.

SAVE_VIBRONIC_PARAMS
       Save information about excited state which is requested in vibronic spectra simulation.
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       FALSE
RECOMMENDATION:
       TRUE

VIBRONIC_SPECTRA
       Specifies which type of vibronic spectra will be predicted. Should be used in a frequency job (jobtype = Freq).
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 No vibronic spectra is predicted. 1 OPA spectra is calculated. 2 OPE spectra is calculated. 3 RRS spectra is calculated.
RECOMMENDATION:
       Use the default.

Variables in the $vibronic section:

MODEL
       Specifies which kind of model will be used to simulate the vibronic spectra.
INPUT SECTION: $vibronic
TYPE:
       INTEGER
DEFAULT:
       -1
OPTIONS:
       1 FC. 2 FCHT. 3 VG.
RECOMMENDATION:
       User defined

TEMPERATURE
       Specifies the temperature in the vibronic spectra simulation.
INPUT SECTION: $vibronic
TYPE:
       FLOAT
DEFAULT:
       298.15
OPTIONS:
       t temperature, in K.
RECOMMENDATION:
       User defined

FREQ_RANGE
       Specifies the frequency range of vibronic spectra.
INPUT SECTION: $vibronic
TYPE:
       FLOAT
DEFAULT:
       1.0 40000.0 10.0
OPTIONS:
       νminνmaxδν minimum, maximum and step size, in cm-1.
RECOMMENDATION:
       User defined

TIME_RANGE
       Specifies the step size and the number of steps in time domain propagation.
INPUT SECTION: $vibronic
TYPE:
       FLOAT and INTEGER
DEFAULT:
       1.0 40000
OPTIONS:
       δtnstep time step size in a.u., and the number of steps.
RECOMMENDATION:
       User defined

DAMPING
       Specifies the damping factor.
INPUT SECTION: $vibronic
TYPE:
       FLOAT
DEFAULT:
       300.0
OPTIONS:
       γ damping factor, in cm-1.
RECOMMENDATION:
       User defined

FREQ_SCALE_FACTOR
       Specifies the frequency scale factors.
INPUT SECTION: $vibronic
TYPE:
       FLOAT
DEFAULT:
       1.0 1.0 1.0 1.0
OPTIONS:
       λHgλHeλZPEgλZPEe scale factor for ground state harmonic frequency, for excited state harmonic frequency, for ground state zero-point energy, and for excited state zero-point energy
RECOMMENDATION:
       User defined

EPSILON
       Specifies the spectral broadening factor. It is available only for RRS spectra simulation.
INPUT SECTION: $vibronic
TYPE:
       FLOAT
DEFAULT:
       25.0
OPTIONS:
       ε broadening factor, in cm-1.
RECOMMENDATION:
       User defined

10.9.5.3 Vibronic Job Examples

Example 10.22  Input files for OPA spectra in the FCHT approximation of formaldehyde corresponding to the S0S1 transition. In the first step, frequency analysis at the S1 equilibrium geometry. Then run a ground state frequency analysis on the S0 ground state optimized structure.

$molecule
  0 1
  O    -0.0367447359    -0.0007590817     0.6963163574
  C     0.1461299638     0.0026846285    -0.5839700302
  H    -0.0732270514     0.9340547891    -1.1138640182
  H    -0.0391581765    -0.9359803358    -1.1140167891
$end

$rem
  JOBTYPE                freq
  METHOD                 b3lyp
  BASIS                  def2-TZVP
  CIS_STATE_DERIV        1
  CIS_SINGLETS           true
  CIS_TRIPLETS           false
  CIS_N_ROOTS            10
  SYM_IGNORE             true
  SAVE_VIBRONIC_PARAMS   true ! enables saving information of S1 state
$end

@@@

$molecule
   0 1
   O    0.0000000000     0.0000000000     0.6637077571
   C    0.0000000000     0.0000000000    -0.5351027012
   H    0.0000000000     0.9394749352    -1.1220697679
   H    0.0000000000    -0.9394749352    -1.1220697679
$end

$rem
   JOBTYPE            freq
   METHOD             b3lyp
   BASIS              def2-TZVP
   SYM_IGNORE         true
   VIBRONIC_SPECTRA   1 !enables vibronic_spectra and reads saved information
$end

$vibronic
   model        2
   freq_range   20000. 60000. 10.
   time_range   1. 40000
   damping      40.
$end

View output

Example 10.23  Vibrationally resolved fluorescence, i.e., OPE with the first kind of FC model, is calculated as following. The emission from D1 to D0 of p-fluorobenzyl radical is used as an example. This is the first job of the total vibronic spectra simulation, by running the excited state geometry optimization retaining information in $QCSCRATCH/savename. Information from this job will be needed to complete the simulation in Example 10.9.5.3.

$molecule
   0 2
   C         1.4840482200    0.0000338155    0.0000000000
   C         0.7160497031    0.0000524901   -1.2119311870
   C        -0.7159596058    0.0000542629   -1.2126930961
   C        -1.4043236629    0.0000543088    0.0000000000
   C        -0.7159596058    0.0000542629    1.2126930961
   C         0.7160497031    0.0000524901    1.2119311870
   C         2.8748450131    0.0000120580    0.0000000000
   H         1.2370230923    0.0000896598   -2.1693859212
   H        -1.2717579173    0.0000412967   -2.1497435961
   H        -1.2717579173    0.0000412967    2.1497435961
   H         1.2370230923    0.0000896598    2.1693859212
   H         3.4346492051    0.0000003003   -0.9330758768
   H         3.4346492051    0.0000003003    0.9330758768
   F        -2.7508602624    0.0000394216    0.0000000000
$end

$rem
   JOBTYPE                opt
   METHOD                 b3lyp
   BASIS                  def2-SVP
   CIS_STATE_DERIV        1
   CIS_N_ROOTS            10
   SYM_IGNORE             true
   SAVE_VIBRONIC_PARAMS   true !saved into $QCSCRATCH/savename
$end

View output

Example 10.24  The final job for obtaining vibrationally resolved fluorescence of the D1 to D0 transition of p-fluorobenzyl radical using OPE with the first kind of FC model from Example 10.9.5.3.

$molecule
   0 2
   C       1.4578807306     0.0130092784     0.0000000000
   C       0.7102753558     0.0082793447    -1.2194714816
   C      -0.6772053823    -0.0007923729    -1.2210832164
   C      -1.3603507249    -0.0052928605     0.0000000000
   C      -0.6772053834    -0.0007923730     1.2210832170
   C       0.7102753546     0.0082793447     1.2194714789
   C       2.8669152234     0.0219746232     0.0000000000
   H       1.2502372081     0.0119050313    -2.1697312391
   H      -1.2498963812    -0.0045285858    -2.1510495277
   H      -1.2498963822    -0.0045285858     2.1510495285
   H       1.2502372136     0.0119050313     2.1697312323
   H       3.4299577819     0.0255536038    -0.9358170827
   H       3.4299577814     0.0255536038     0.9358170860
   F      -2.7010161586    -0.0142661666     0.0000000000
$end

$rem
   JOBTYPE            freq
   METHOD             b3lyp
   BASIS              def2-SVP
   SYM_IGNORE         true
   VIBRONIC_SPECTRA   2
$end

$vibronic
   model        1
   temperature  0.
   freq_range   1. 40000. 10.
   time_range   1. 40000
   damping      20.
$end

View output

Example 10.25  RRS spectra of phenoxyl radical (D0D3 transition) with the VG approximation. Therefore the first job calculates the D3 state force at the ground state optimized geometry, followed by the ground state frequency analysis. The excited state forces and ground state frequencies are calculated in the ground state equilibrium geometry.

$molecule
   0 2
   C       0.0000000000     1.2271514002    -1.0879472096
   C       0.0000000000     0.0000408897    -1.7873074655
   C       0.0000000000    -1.2270324440    -1.0880160727
   C       0.0000000000    -1.2409681161     0.2924435676
   C       0.0000000000    -0.0000313560     1.0551142042
   C       0.0000000000     1.2409428316     0.2924458686
   H       0.0000000000     2.1656442172    -1.6487551860
   H       0.0000000000    -0.0001767539    -2.8803293768
   H       0.0000000000    -2.1655968771    -1.6487220344
   H       0.0000000000    -2.1715667156     0.8648121894
   H       0.0000000000     2.1714692701     0.8649813475
   O       0.0000000000     0.0001236541     2.3063351676
$end

$rem
   JOBTYPE                force
   METHOD                 b3lyp
   BASIS                  def2-SVP
   CIS_STATE_DERIV        3
   CIS_N_ROOTS            10
   SYM_IGNORE             true
   SAVE_VIBRONIC_PARAMS   true
$end

@@@

$molecule
   read !VG approximation uses the same geometry for ground and excited state
$end

$rem
   JOBTYPE            freq
   METHOD             b3lyp
   BASIS              def2-SVP
   SYM_IGNORE         true
   VIBRONIC_SPECTRA   3
$end

$vibronic
   MODEL         3
   TEMPERATURE   0.
   FREQ_RANGE    1. 4000. 1.
   TIME_RANGE    1. 40000
   DAMPING       100.
   EPSILON       25.
$end

View output