Vibrational Spectroscopy¶

Where are the results?¶

Because the results of a vibrational spectroscopy calculation are tied to a particular point on the potential energy surface, they are found on the engine output files. Note also that the properties are not always calculated in every PES point that the AMS driver visits during a calculation. By default they are only calculated in special PES points, where the definition of special depends on the task AMS is performing: For a geometry optimization properties would for example only be calculated at the final, converged geometry. This behavior can often be modified by keywords special to the particular running task.

All vibrational Modes¶

The calculation of the normal modes of vibration can be requested with:

Properties
   NormalModes Yes/No
End

Typically used icw with Task SinglePoint, Task GeometryOptimization, or Task TransitionStateSearch. In case of geometry optimization or transition state search the normal modes will only be calculated at the final, converged geometry.

Properties

NormalModes

Type:	Bool
Default value:	No
GUI name:	Frequencies
Description:	Calculate the frequencies and normal modes of vibration, and for molecules also the corresponding IR intensities if the engine supports the calculation of dipole moments.

The molecular normal modes are normally calculated within the harmonic oscillation model. If the molecule is in its equilibrium conformation, it sits in the lowest point (at least locally) on the PES. The cross-section of the PES profile close to this point can then be assumed to be approximately parabolic, such that the second derivative of the energy w.r.t a nuclear coordinate can be interpreted as a force constant for the harmonic oscillation of an atom along this coordinate. Since molecular vibrations in polyatomics involve the simultaneous displacement of multiple atoms, this harmonic oscillator model can be generalized to multiple nuclear coordinates. The normal modes and their frequencies then become eigenvectors and eigenvalues of a force constant matrix, the Hessian:

The (non-mass-weighted) Hessian is saved in the engine result file as variable AMSResults%Hessian. It is not printed to the text output. The column/row indices are ordered as: x-component of atom 1, y-component of atom 1, z-component of atom 1, x-component of atom 2, etc.

Most engines cannot calculate the Hessian analytically. The Hessian is then constructed column-wise through numerical differentiation of the energy gradients w.r.t. each nuclear coordinate. AMS will set up 2 single-point calculations (1 for the positive displacement, 1 for the negative displacement), and the requested engine will return the energy gradients at these displacements. These gradients are calculated analytically for most engines.

When requesting the normal modes calculation, integrated IR intensities are simultaneously calculated during the finite differentiation steps when constructing the Hessian (as long as dipole moments are supported by the engine). These IR intensities are calculated from the numerical dipole gradients:

Where \(\alpha\) denotes the x-,y- and z-components of the dipole moment \(\mu\), and \(Q^m\) is the mass-weighted vibrational normal mode.

The resulting IR spectrum can be visualized by opening the engine result file with AMSspectra. The normal modes of vibration and the IR intensities are saved to the engine result file in the Vibrations section.

Symmetry labels of the normal modes may be calculated if AMS uses symmetry in the calculation (key UseSymmetry). If symmetry is used, the normal modes are projected against symmetric displacements for each irrep. If that is not successful the symmetry label is ‘MIX’. Symmetry is only recognized if the geometry is (almost) perfectly symmetric and has a specific orientation in space. You can use the Symmetrize key in the System block to symmetrize and reorient the molecule. If the AMSinput GUI module is used, one can click the Symmetrize button (the star) and the GUI will try to symmetrize and reorient the molecule.

NormalModes
   ReScanModes Yes/No
   ReScanFreqRange float_list
End

Symmetric Displacements¶

NormalModes
   Displacements Symmetric
End

Specify Displacements Symmetric to calculate the energy Hessian using finite differences in symmetry-adapted displacements, and the corresponding normal modes.

NormalModes
   SymmetricDisplacements
      Type [All | Infrared | Raman | InfraredAndRaman]
   End
End

If Type InfraRed or Type Raman is specified then only irreps that result in non-zero intensities for the corresponding spectroscopy will be included in the calculation. Using this feature may save a lot of time for large symmetric molecules by skipping calculation of normal modes that would not contribute to the spectrum anyway. If Type InfraRedAndRaman is specified then vibrations that have a non-zero IR or Raman intensity will be calculated. If Type All is specified then all vibrations will be calculated. For multi-dimensional irreps (such as E and T) only the first component will be computed. For any component beyond the first, the frequencies and intensities will be copied from the first one.

NormalModes

SymmetricDisplacements

Type:	Block
Description:	Configures details of the calculation of the frequencies and normal modes of vibration in symmetric displacements.

Type

Type:	Multiple Choice
Default value:	All
Options:	[All, Infrared, Raman, InfraredAndRaman]
GUI name:	Symm Frequencies
Description:	For symmetric molecules it is possible to choose only the modes that have non-zero IR or Raman intensity (or either of them) by symmetry. In order to calculate the Raman intensities the Raman property must be requested.

Mobile Block Hessian (MBH)¶

NormalModes
   Displacements Block
End

Specify Displacements Block for the Block Normal Modes option (also known as Mobile Block Hessian, or MBH [1] [2]). MBH is useful when calculating vibrational frequencies of a small part of a very large system (molecule or cluster). Calculation of the full spectrum of such a system may be inefficient and is unnecessary if one is interested in one particular part. Besides, it may be difficult to extract normal modes related to the interesting sub-system out of the whole spectrum. Using Block Normal Modes it is possible to treat parts of the system as rigid blocks. Each block will usually have only six frequencies related to its rigid motions compared to 3*N for when each atom of the block is treated separately.

MBH is suitable to calculate frequencies in partially optimized structures. Assume a geometry optimization is performed with the Block key in the Constraints input block [see constrained geometry optimizations]. During the geometry optimization, the shape of the block is not changed. The internal geometry of the block is kept fixed, but the block as a whole can still translate or rotate.

At the end of such a partial geometry optimization, the position and orientation of the block is optimized, thus the total force on the block is zero. However, there might be still some residual forces within a block, since those degrees of freedom were not optimized. A traditional frequency calculation performed on this partially optimized structure might result in non-physical imaginary frequencies without a clear interpretation. Therefore one should use an adapted formulation of normal mode analysis: the Mobile Block Hessian method. MBH does not consider the internal degrees of freedom of the block (on which residual forces) apply, but instead uses the position/orientation of the block as coordinates. In the resulting normal mode eigenvectors, all atoms within the same block move collectively.

Of course, MBH can also be applied on a fully optimized structure.

Accuracy

NormalModes
   BlockDisplacements
      AngularDisplacement float
      BlockAtoms integer_list
      BlockRegion string
      Parallel
         nCoresPerGroup integer
         nGroups integer
         nNodesPerGroup integer
      End
      RadialDisplacement float
   End
End

NormalModes

BlockDisplacements

Type:	Block
Description:	Configures details of a Block Normal Modes (a.k.a. Mobile Block Hessian, or MBH) calculation.

AngularDisplacement

Type:	Float
Default value:	0.5
Unit:	Degree
Description:	Relative step size for rotational degrees of freedom during Block Normal Modes finite difference calculations. It will be scaled with the characteristic block size.

BlockAtoms

Type:	Integer List
Recurring:	True
Description:	List of atoms belonging to a block. You can have multiple BlockAtoms.

BlockRegion

Type:	String
Recurring:	True
Description:	The region to to be considered a block. You can have multiple BlockRegions, also in combination with BlockAtoms.

Parallel

Type:	Block
Description:	Configuration for how the individual displacements are calculated in parallel.

nCoresPerGroup

Type:	Integer
Description:	Number of cores in each working group.

nGroups

Type:	Integer
Description:	Total number of processor groups. This is the number of tasks that will be executed in parallel.

nNodesPerGroup

Type:	Integer
GUI name:	Cores per task
Description:	Number of nodes in each group. This option should only be used on homogeneous compute clusters, where all used compute nodes have the same number of processor cores.

RadialDisplacement

Type:	Float
Default value:	0.005
Unit:	Angstrom
Description:	Step size for translational degrees of freedom during Block Normal Modes finite difference calculations.

The second derivatives of the energy with respect to Cartesian displacements of the free atoms and those with respect to block motions (3 translation plus 3 rotations) are calculated by numerical differentiation of the gradient. The accuracy of the second derivatives is determined by the accuracy of the gradient evaluation and the step size in the numerical differentiation. The RadialDisplacement and AngularDisplacement parameters can be specified to set the step size for Cartesian displacements (translations) and block rotations respectively. The step size for angles is automatically scaled with the block size.

Mode Scanning¶

Mode Scanning can be used to obtain more accurate approximations for properties obtained by numerical differentiation along the vibrational normal modes (frequencies, intensities, Raman, etc.), without changing the modes themselves. Mode Scanning is an extension of the frequency scanning options (ScanFreq) that were part of ADF and BAND in earlier versions of the Amsterdam Modeling Suite. These latter options are still available as the ReScanModes keyword in the NormalModes block, if these are requested during a calculation.

• Primarily used to identify spurious imaginary modes.
• Improve numerical accuracy of normal mode properties.
• Rescanning modes using a different level of theory.

Task VibrationalAnalysis
VibrationalAnalysis
   Type ModeScanning
   Displacement 0.001
   NormalModes
     ModeFile adf.rkf
     # select all modes with imaginary frequencies
     ModeSelect
        ImFreq true
     End
   End
End

Mode Refinement¶

With this option you can improve the normal modes, by importing previously calculated modes and then applying a more accurate engine, or more accurate settings, typically for only part of the spectrum. The vibrational Mode Refinement method not only refines frequencies from a previous calculation, but also tries to correct the vibrational modes themselves.

• Refinement of spectral regions requires a sufficient number of modes in the basis to be accurate.
• 1-step refinement. No iterative improvement possible. (Unless followed by a separate Mode Tracking calculation.)
• Quality of the results depends on accuracy of the selected guess modes.

If we start from e.g. a semi-empirical method such as in MOPAC, we can get approximations for the vibrational modes. Mode Refinement then re-calculates part of the Hessian for a subset of these modes using a more accurate method such as GGA DFT, and updates the normal modes themselves to fit this more accurate method. It is intended to circumvent the expensive calculation of the Hessian if you are only interested in a (small) part of the full spectrum. This is based on the method in reference [3].

Because the Mode Refinement method uses linear combinations of the guess modes, its accuracy depends on the set of modes that is supplied.

• If we want to e.g. obtain a mode which includes a C=O stretch, then the initial set must contain a mode which has this C=O stretch, otherwise this cannot be included in the refined modes.
• If we are refining a region containing many similar modes, e.g. vibrations of aromatic ring backbones, and we only use part of this spectral region for the initial set, the set of refined modes will “drift” towards the centre of the spectral region as a results of mode-mixing. This is again an artefact of missing character in the modes.
• This mode-mixing may result in reduced accuracy for some of the modes, as this procedure minimizes the total error for all of the modes. Instead of having a couple of modes with large errors, mode-mixing tends to spread out the error across multiple normal modes. Adding 1 “bad” mode to the basis can then negatively affect your results.
• The advantage of Mode Refinement over Mode Tracking is the ability to refine entire spectral regions at once. If we have a good basis, Mode Refinement can be less computationally expensive than Mode Tracking. If you want to refine larger sections of the spectrum, Mode Refinement is therefore recommended. If you only want to calculate a select few modes, use Mode Tracking to avoid basis dependence and to assure accuracy of the obtained modes.
• For characteristic peaks, Mode Tracking shows very good convergence, and will thus be cheaper to use than Mode Refinement. For (semi-)degenerate modes however, Mode Refinement works better due to the poor tracking performance for these modes.

Task VibrationalAnalysis
VibrationalAnalysis
   Type ModeRefinement
   Displacement 0.001
   NormalModes
     ModeFile adf.rkf
     ModeSelect
        ...
     End
     ScanModes true
   End
End

Mode Tracking¶

The Mode Tracking task is an interface for mode- and intensity-tracking methods, adapted from the MoViPac suite [4]- [5]. These methods can be used to obtain select normal modes, without having to calculate the entire vibrational spectrum. It does this through an iterative procedure.

• Calculations are conducted for each mode separately. Converges fastest for characteristic (non-highly degenerate) modes.
• Iterative approximation to the true modes. Guaranteed to give the correct normal modes if the procedure converges.
• Will not necessarily reproduce the entire spectrum as multiple guess modes can converge to the same normal mode.

Mode Tracking uses information about the known parts of the Hessian to expand its basis iteratively:

• Missing C-O stretch character can thus be recovered in this procedure, and there is no basis dependency.
• For large regions with similar modes however, it is possible that multiple guess modes converge to the same normal mode. Running mode tracking for all modes in this region might not reproduce all unique normal modes.
• The advantage of Mode Refinement over Mode Tracking is the ability to refine entire spectral regions at once. If we have a good basis, Mode Refinement can be less computationally expensive than Mode Tracking. If you want to refine larger sections of the spectrum, Mode Refinement is therefore recommended. If you only want to calculate a select few modes, use Mode Tracking to avoid basis dependence and to assure accuracy of the obtained modes.
• For characteristic peaks, Mode Tracking shows very good convergence, and will thus be cheaper to use than Mode Refinement. For (semi-)degenerate modes however, Mode Refinement works better due to the poor tracking performance for these modes.

Mode Tracking starts with a numerical frequency calculation, which refines the initial guess \(b^m\) for the selected mode. The error of this mode with respect to the true Hessian eigenvector is calculated. This error is used in a (Jacobi-)Davidson algorithm to generate an additional mode. In subsequent iterations, we use these modes as approximations to the true normal modes. In this way, the error of the mode is minimized iteratively, yielding a closer approximation to true normal modes. This is how Mode Tracking differs from the Mode Refinement methods, in that it guarantees that the obtained modes are correct (assuming the procedure has converged).

Task VibrationalAnalysis
VibrationalAnalysis
   Type ModeTracking
   ...
   ModeTracking
      HessianGuess [Unit | File | UFF | Inline]
      HessianInline # Non-standard block. See details.
         ...
      End
      HessianPath string
      UpdateMethod [JD | D | I]
      MaxIterations integer
      ...
      GramSchmidt [True | False]
      GramSchmidtIterations integer
      GramSchmidtTolerance float
   End
End

Task VibrationalAnalysis
VibrationalAnalysis
   Type ModeTracking
   Displacement float
   ...
   NormalModes
      ScanModes [True | False]
      ...
   End
End

Task VibrationalAnalysis
VibrationalAnalysis
   Type ModeTracking
   ModeTracking
      TrackingMethod [OverlapInitial, DifferenceInitial, FreqInitial, IRInitial,
                      OverlapPrevious, DifferencePrevious, FreqPrevious, IRPrevious,
                      HighestFreq, HighestIR, LowestFreq, LowestResidual]
      ...
   End
End

Task VibrationalAnalysis
VibrationalAnalysis
   Type ModeTracking
   NormalModes
      ModeInputFormat [File | Inline | Hessian]
      ModeFile string
      ModeInline # Non-standard block. See details.
         ...
      End
      ModeSelect
         ...
      End
      MassWeighInlineMode [True | False]
  End
End

Task VibrationalAnalysis
VibrationalAnalysis
   Type ModeTracking
   ModeTracking
      ToleranceForNorm float
      ToleranceForResidual float
      ToleranceForBasis float
   End
End

Selecting modes¶

Mode Scanning, Mode Refinement and Mode Tracking as well as VG-FC Vibronic-Structure, VG-FC Vibronic-Structure Refinement and VG-FC resonance Raman all require a set of normal modes to operate on. For Mode Scanning these are the modes that you want to calculate the properties of, for Mode Refinement these modes form the basis modes, and for Mode Tracking these are the initial guess modes. For the VG-FC based methods these modes are the modes responsible for the vibronic coupling to the electronic excitation (in VG-FC Vibronic-Structure Refinement they are refined first).

These methods provide options to load a large set of modes, after which the program will filter out the modes of interest. This is done according to the keys set in the ModeSelect block.

Below is an overview of all the available options for the ModeSelect block as they appear in the basic vibrational analysis tools. The vibronic variants are discussed in more detail on their respective documentation pages.

The options below are not mutually exclusive.

ModeSelect
   DisplacementBound float
   FreqAndIRRange float_list
   FreqRange float_list
   Full Yes/No
   HighFreq integer
   HighIR integer
   IRRange float_list
   ImFreq Yes/No
   LargestDisplacement integer
   LowFreq integer
   LowFreqNoIm integer
   LowIR integer
   ModeNumber integer_list
End

ModeSelect

DisplacementBound

Type:	Float
Description:	Vibronic Structure (Refinement), Resonance Raman: Select all modes with a dimensionless oscillator displacement greater than the specified value.

FreqAndIRRange

Type:	Float List
Unit:	cm-1 and km/mol
Recurring:	True
Description:	Specifies a combined frequency and IR intensity range within which all modes will be selected. First 2 numbers are the frequency range, last 2 numbers are the IR intensity range.

FreqRange

Type:	Float List
Unit:	cm-1
Recurring:	True
Description:	Specifies a frequency range within which all modes will be selected. 2 numbers: an upper and a lower bound. Calculating all modes higher than some frequency can be achieved by making the upper bound very large.

Full

Type:	Bool
Default value:	No
GUI name:	All modes
Description:	Select all modes. This only make sense for Mode Scanning calculations.

HighFreq

Type:	Integer
GUI name:	# High frequencies
Description:	Select the N modes with the highest frequencies.

HighIR

Type:	Integer
GUI name:	# High IR
Description:	Select the N modes with the largest IR intensities.

IRRange

Type:	Float List
Unit:	km/mol
Recurring:	True
Description:	Specifies an IR intensity range within which all modes will be selected. 2 numbers: an upper and a lower bound.

ImFreq

Type:	Bool
Default value:	No
GUI name:	All imaginary frequencies
Description:	Select all modes with imaginary frequencies.

LargestDisplacement

Type:	Integer
Description:	Vibronic Structure (Refinement), Resonance Raman: Select the N modes with the largest VG-FC displacement.

LowFreq

Type:	Integer
GUI name:	# Low frequencies
Description:	Select the N modes with the lowest frequencies. Includes imaginary modes which are recorded with negative frequencies.

LowFreqNoIm

Type:	Integer
GUI name:	# Low positive frequencies
Description:	Select the N modes with the lowest non-negative frequencies. Imaginary modes have negative frequencies and are thus omitted here.

LowIR

Type:	Integer
GUI name:	# Low IR
Description:	Select the N modes with the smallest IR intensities.

ModeNumber

Type:	Integer List
GUI name:	Mode numbers
Description:	Indices of the modes to select.

Thermodynamics (ideal gas)¶

The following thermodynamic properties are calculated by default whenever normal modes are computed: entropy, internal energy, constant volume heat capacity, enthalpy and Gibbs free energy. Translational, rotational and vibrational contributions are calculated for entropy, internal energy and constant volume heat capacity.

The results are written to the output file (section: “Statistical Thermal Analysis”) and to the engine binary results file (section: “Thermodynamics”).

The thermodynamic properties are computed assuming an ideal gas, and electronic contributions are ignored. The latter is a serious omission if the electronic configuration is (almost) degenerate, but the effect is small whenever the energy difference with the next state is large compared to the vibrational frequencies. The thermal analysis is based on the temperature dependent partition function. The energy of a (non-linear) molecule is (if the energy is measured from the zero-point energy)

The summation is over all harmonic \(\nu_j\), \(h\) is Planck’s constant and \(D\) is the dissociation energy

Contributions from low (less than 20 1/cm) frequencies to entropy, heat capacity and internal energy are excluded from the total values, but they are listed separately (so the user can add them if they wish).

As an alternative to outright excluding low-frequency contributions, a correction scheme is available that is based on interpolating between harmonic oscillator and free rotor values [7] [8] (Li/Head-Gordon and Grimme). It can greatly reduce the impact of the inaccuracies of the harmonic oscillator model on thermodynamic properties at these low frequencies. The scheme corrects vibrational contributions to entropies, internal energies and constant volume heat capacities. This correction is applied automatically and its results are printed separately (in the text output, the corrected terms are marked with the symbol (c)). When applied, the correction considers all real frequencies, including those less than 20 1/cm.

The interpolation for a corrected thermodynamic property \(f\) at pressure \(p\) and temperature \(T\) in terms of harmonic oscillator terms \(f_{HO}\), free rotor terms \(f_{FR}\), and interpolator terms \(x\) for each harmonic oscillator frequency \(\nu_j\) is:

Where \(\alpha\) is an arbitrary exponent and \(\nu_0\) is the harmonic oscillator frequency around which \(x\) interpolates, with \(x=0.5\) when \(\nu_j=\nu_0\) and \(x\approx1.0\) when \(\nu_j\gg\nu_0\). While the free rotor terms used for internal energies and heat capacities are the standard ones, the terms used for entropies have to use the rotors’ moments of inertia \(\mu_{FR}\) and symmetry \(\sigma\), which formally cannot be calculated from harmonic frequencies alone. The correction scheme instead estimates each moment of inertia as being of a \(\sigma=1\) free rotor whose first excited state has an energy equal to the given \(h\nu_{j}\). After this, each obtained moment of inertia \(\mu_{FR}\) is modified by an averaging moment of inertia \(\mu_{av}\) to avoid grossly overestimating entropies at very small frequencies (less than around 1 1/cm):

Input options

Thermo
   Temperatures float_list
   Pressure float
   LowFrequencyCorrector
      Alpha float
      Frequency float
      MomentOfInertia float
   End
End

Thermo

Type:	Block
Description:	Options for thermodynamic properties (assuming an ideal gas). The properties are computed for all specified temperatures.

Temperatures

Type:	Float List
Default value:	[298.15]
Unit:	Kelvin
Description:	List of temperatures at which the thermodynamic properties will be calculated.

Pressure

Type:	Float
Default value:	1.0
Unit:	atm
Description:	The pressure at which the thermodynamic properties are computed.

LowFrequencyCorrector

Type:	Block
Description:	Options for the dampener-powered free rotor interpolator that corrects thermodynamic quantities for low frequencies. See DOI:10.1021/jp509921r and DOI:10.1002/chem.201200497.

Alpha

Type:	Float
Default value:	4.0
Description:	The exponent term used in the dampener.

Frequency

Type:	Float
Default value:	100.0
Unit:	cm-1
Description:	The frequency around which the dampener interpolates between harmonic oscillator and free rotor quantities.

MomentOfInertia

Type:	Float
Default value:	1e-44
Unit:	kg m^2
GUI name:	Averaging Moment of Inertia
Description:	The moment of inertia used to restrict entropy results for very small frequencies (generally around less than 1 cm-1).

Zero-point energy (Hartree):     0.0333

...
...

Temp                                                        Transl      Rotat     Vibrat     Total
----                                                        ------      -----     ------     -----

298.15   Entropy (cal/mol-K):                              34.441     11.474      0.137     46.052
         Nuclear Internal Energy (kcal/mol):                0.889      0.889     20.941     22.718
         Constant Volume Heat Capacity (cal/mol-K):         2.981      2.981      0.565      6.526

 Summary of energy terms
                                              hartree              eV         kcal/mol           kJ/mol
                                 --------------------     -----------       ----------      -----------
 Energy from Engine:               -0.743995039793930        -20.2451          -466.86         -1953.36
 Nuclear Internal Energy:           0.036203917534227          0.9852            22.72            95.05
 Internal Energy U:                -0.707791122259703        -19.2599          -444.14         -1858.31
 pV/n = RT:                         0.000944186013486          0.0257             0.59             2.48
 Enthalpy H:                       -0.706846936246217        -19.2343          -443.55         -1855.83
 -T*S:                             -0.021880868282982         -0.5954           -13.73           -57.45
 Gibbs free energy:                -0.728727804529199        -19.8297          -457.28         -1913.27

Moments of inertia¶

In case normal modes are computed in AMS, AMS also reports the moments of inertia of the molecule in units of amu bohr² (amu = atomic mass unit) and its corresponding principal axes.

Partial Vibrational Spectra (PVDOS)¶

The Partial Vibrational Spectra (also known as PVDOS) is computed by default whenever normal modes are requested. The PVDOS \(P_{I,n}\) for atom \(I\) and normal mode \(n\) is defined as:

where \(m_I\) is the nuclear weight of atom \(I\), and \(\vec{\eta}_{I,n}\) is the displacement vector for atom \(I\) in normal normal mode \(n\).

Fig. 2 Example of partial vibrational spectrum (PVDOS). The dotted line is the full IR spectrum of 1-propanol. The solid line is the PVDOS-scaled IR spectrum of the OH group (IR spectrum computed using GFN1-xTB).

The PVDOS matrix is not printed to the text output, but only saved to the engine binary output (.rkf) in the variable Vibrations%PVDOS.

Raman¶

In this method the Raman scattering spectrum is calculated from the geometrical derivatives of the frequency-dependent polarizability. Engine ADF is required. Raman scattering intensities and depolarization ratios for all or a selected number of molecular vibrations at a certain laser frequency can be calculated. The Raman scattering calculation is very similar to an IR intensity calculation. In fact, all IR output is automatically generated as well. At all distorted geometries the dipole polarizability tensor is calculated. This is time-consuming and is only feasible for small molecules.

Properties
   Raman Yes/No
End

Raman
   IncidentFrequency float
   FreqRange float_list
End

If a FreqRange is included the Raman intensities are calculated for a range of vibrational frequencies only. Using this option is a fast alternative for calculating all Raman intensities.

Properties

Raman

Type:	Bool
Default value:	No
Description:	Requests calculation of Raman intensities for vibrational normal modes.

Raman

IncidentFrequency

Type:	Float
Default value:	0.0
Unit:	eV
Description:	Frequency of incident light.

FreqRange

Type:	Float List
Unit:	cm-1
Recurring:	True
GUI name:	Frequency range
Description:	Specifies a frequency range within which all modes will be scanned. 2 numbers: an upper and a lower bound.

Resonance Raman: excited-state finite lifetime¶

Resonance Raman spectroscopy uses incident light with a wavelength close to that of an electronic transition. In this method (Ref. [11]) the resonance Raman-scattering (RRS) spectra is calculated from the geometrical derivatives of the frequency-dependent polarizability. Engine ADF is required. The polarizability derivatives are calculated from resonance polarizabilities by including a finite lifetime (phenomenological parameter) of the electronic excited states.

Raman
   FreqRange float_list
   IncidentFrequency float
   LifeTime float
End

Raman

FreqRange

Type:	Float List
Unit:	cm-1
Recurring:	True
GUI name:	Frequency range
Description:	Specifies a frequency range within which all modes will be scanned. 2 numbers: an upper and a lower bound.

IncidentFrequency

Type:	Float
Default value:	0.0
Unit:	eV
Description:	Frequency of incident light.

LifeTime

Type:	Float
Default value:	0.0
Unit:	hartree
Description:	Specify the resonance peak width (damping) in Hartree units. Typically the lifetime of the excited states is approximated with a common phenomenological damping parameter. Values are best obtained by fitting absorption data for the molecule, however, the values do not vary a lot between similar molecules, so it is not hard to estimate values. A typical value is 0.004 Hartree.

It is similar to the simple excited-state gradient approximation method (see next section) if only one electronic excited state is important, however, it is not restricted to only one electronic excited state. In the limit that there is only one possible state in resonance the two methods should give more or less the same results. However, for many states and high-energy states and to get resonance Raman profiles (i.e., Raman intensities as a function of the energy of the incident light beam) this approach might be more suitable. The resonance Raman profiles in this approach are averaged profiles since vibronic coupling effects are not accounted for.

Resonance Raman: VG-FC¶

According to a the time-dependent picture of resonance-Raman (RR) scattering the relative intensities of RR scattering cross sections are, under certain assumptions, proportional to the square of the excited-state energy gradients projected onto the ground-state normal modes of the molecule (see Ref. [12]). For an alternative implementation of RR scattering using a finite lifetime of the excited states, and a discussion of some of the differences, see the previous section. Engine ADF or DFTB is required.

The vertical gradient Franck-Condon (VG-FC) method, also called the Independent Mode Displaced Harmonic Oscillator (IMDHO) model, we use to calculate vibrationally resolved absorption spectra can also be applied to the calculation of resonance Raman spectra. In resonance Raman spectroscopy a molecule is excited from its ground state to some electronically excited state. After a short period of time, the molecule then relaxes back to its electronic ground state. However, when doing so, it might end up in a different vibrational state than it started off in. The result is an energy difference between the incident and emmitted photon. One can then plot the intensity for different energy differences to produce what is known as a Raman spectra. Resonance Raman spectroscopy uses incident light with a wavelength close to that of an electronic transition.

AMS supports the calculation of such spectra by modeling the vibronic coupling of electronic transitions using the VG-FC model. This model is discussed also on the Vibronic-Structure documentation page. Here we will discuss the modifications necessary to use the VG-FC model for resonance Raman spectroscopy. It is worth noting that this VG-FC resonance Raman application does not support the mode selective options. As a result the VG-FC Resonance Raman application will always first perform a full frequency analysis to obtain its normal modes.

Task VibrationalAnalysis
VibrationalAnalysis
   Task ResonanceRaman
   ResonanceRaman
      IncidentFrequency float
      LifeTime float
      RamanOrder integer
      RamanRange float_list
      MaximumStates integer
   End
   ...
End

VibrationalAnalysis
   Type ResonanceRaman
   NormalModes
     ModeSelect
       Full True
     End
   End
   ExcitationSettings
     ExcitationInfo File
     ExcitationFile ./your_excitation/dftb.rkf
     Singlet
       A 1 2 4
     End
   End
   ResonanceRaman
     RamanOrder 3
     RamanRange 0.0 3000.0
   End
End

Engine ADF¶

In the ADF engine a method is implemented to calculate both on- and off-resonance vibrational Raman optical activities (VROAs) of molecules using time-dependent density functional theory, see Ref. [14]. This is an extension of a method to calculate the normal VROA by including a finite lifetime of the electronic excited states in all calculated properties. The method is based on a short-time approximation to Raman scattering and is, in the off-resonance case, identical to the standard theory of Placzek. The normal and resonance VROA spectra are calculated from geometric derivatives of the different generalized polarizabilities obtained using linear response theory which includes a damping term to account for the finite lifetime. Gauge-origin independent results for normal VROA have been ensured using either the modified-velocity gauge or gauge-included atomic orbitals. In ADF2016 the velocity gauge tensors required for the calculation of VROA are now correctly calculated with the life time damping parameter. With these complex tensors fixed, resonance VROA intensities are now origin invariant in the velocity gauge, see also Ref. [15].

Atomic polar tensor (APT) model¶

In the so-called atomic polar tensor (APT) model the atomic axial tensors (AATs) can be calculated from electric dipole gradients. Note that the APT model may not be very reliable for predicting VCD bands and its implementation should not be blindly applied beyond a quick assessment. Results using the engine DFTB can be found in [16].

For the engines BAND and DFTB only the APT model can be used. In case of the engine ADF the default is to calculate the VCD analytically, see next section. One can calculate VCD using the APT model with the ADF engine if one includes:

NormalModes
  Hessian Numerical
End

Analytical VCD in ADF¶

In the ADF engine the VCD intensities are calculated using Stephens’ equations for VCD. For the calculation of the atomic axial tensors (AATs), analytical derivatives techniques and London atomic orbitals (the so called GIAO) are employed. As a result the calculated rotational strengths are origin independent, and therefore the common origin gauge is used [17].

New in AMS2020 is that one can calculate analytical VCD also for open-shell systems in a spin-unrestricted calculation.

Calculation of the AATs requires an analytical frequencies calculation. This limits the choice of functionals that can be used for VCD calculations.

The accuracy of the vibrational rotational strengths are determined by the accuracy of the harmonic force field, atomic polar tensors (APTs) and AATs. The most critical parameter being the harmonic force field. Thus, for a fair comparison with experimental data, accurate geometries and functionals that yield accurate force fields (e.g. BP86, OLYP, etc) should be used. Our tests showed that the BP86 functional in combination with TZP basis sets is always a safe choice. For a comparison of VCD spectra calculated with various functionals (e.g BP86, OLYP, BLYP, B3PW91 and B3LYP) see [17]. Regarding the geometries, we recommend the following strict settings, 10^-4 for the geometry convergence of the gradients, and BeckeGrid quality good. The default settings should be used for the calculation of the frequencies.

By default, only the vibrational rotational strengths are printed in the AMS output file. For a deeper insight regarding the origin of the VCD intensity of a given normal mode one can use the auxiliary program VCDtools. VCDtools can be used with the ADF-GUI module ADFspectra.