10.7 Harmonic Vibrational Analysis 10.7 Harmonic Vibrational Analysis 10.7.2 Isotopic Substitutions and Changes in

T

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10.7.1 Overview

(September 1, 2024)

Vibrational analysis is an extremely important tool for the quantum chemist, supplying a molecular fingerprint which is invaluable for aiding identification of molecular species in many experimental studies. Q-Chem includes a vibrational analysis package that can calculate vibrational frequencies and their infrared and Raman activities. ⁶⁰⁷ Johnson B. G., Florián J.
Chem. Phys. Lett.
(1995), 247, pp. 120. Link Vibrational frequencies are calculated by either using an analytic Hessian (if available; see Table 9.2) or, numerical finite difference of the gradient. The default setting in Q-Chem is to use the highest analytical derivative order available for the requested theoretical method. The performance of various ab initio theories in determining vibrational frequencies has been well documented. ⁹⁰⁵ Murray C. W. et al.
Chem. Phys. Lett.
(1992), 199, pp. 551. Link ^, ⁶¹⁰ Johnson B. G., Gill P. M. W., Pople J. A.
J. Chem. Phys.
(1993), 98, pp. 5612. Link ^, ¹¹⁴¹ Scott A. P., Radom L.
J. Phys. Chem.
(1996), 100, pp. 16502. Link

When calculating analytic frequencies at the HF and DFT levels of theory, the coupled-perturbed SCF equations must be solved. This is the most time-consuming step in the calculation, and also consumes the most memory. The amount of memory required is ${\cal{O}}({N^{2}M})$ where $N$ is the number of basis functions, and $M$ the number of atoms. This is an order more memory than is required for the SCF calculation, and is often the limiting consideration when treating larger systems analytically. Q-Chem incorporates a new approach to this problem that avoids this memory bottleneck by solving the CPSCF equations in segments. ⁶⁷¹ Korambath P. P. et al.
Mol. Phys.
(2002), 100, pp. 1755. Link Instead of solving for all the perturbations at once, they are divided into several segments, and the CPSCF is applied for one segment at a time, resulting in a memory scaling of ${\cal{O}}({N^{2}M/N_{\mathrm{seg}}})$ , where $N_{\mathrm{seg}}$ is the number of segments. This option is invoked automatically by the program.

After a vibrational analysis, Q-Chem computes useful statistical thermodynamic properties at standard temperature and pressure following the rigid-rotor-harmonic-oscillator (RRHO) approach. These include the zero-point vibration energy (ZPVE) and, translational, rotational and vibrational, entropies and enthalpies. Note: in the Q-Chem output the “total enthalpy” actually means the total enthalpy correction to the internal energy. One must add this “total enthalpy” to the internal energy to obtain the total enthalpy in common sense. In addition to these thermal corrections, Q-Chem also prints the interpolated vibrational entropy and enthalpy according to the quasi-RRHO (qRRHO) approach by Head-Gordon and co-workers, ⁷⁷⁶ Li Y.-P. et al.
J. Phys. Chem. C
(2015), 119, pp. 1840. Link which extends Grimme’s previous scheme ⁴⁶⁰ Grimme S.
Chem. Eur. J
(2012), 18, pp. 9955. Link to address low-frequency vibrations; Section 10.7.3.

10.7.1.1 Job Control

In order to carry out a frequency analysis users must at a minimum provide a molecule within the $molecule keyword and define an appropriate level of theory within the $rem keyword using the $rem variables EXCHANGE, CORRELATION (if required) (Chapter 4) and BASIS (Chapter 8). Since the default type of job (JOBTYPE) is a single point energy (SP) calculation, the JOBTYPE $rem variable must be set to FREQ.

It is very important to note that a vibrational frequency analysis must be performed at a stationary point on the potential surface that has been optimized at the same level of theory. Therefore a vibrational frequency analysis most naturally follows a geometry optimization in the same input deck, where the molecular geometry is obtained (see examples).

Users should also be aware that the quality of the quadrature grid used in DFT calculations is more important when calculating second derivatives. The default grid for some atoms has changed in Q-Chem 3.0 (see Section 5.5) and for this reason vibrational frequencies may vary slightly form previous versions. It is recommended that a grid larger than the default grid is used when performing frequency calculations.

The standard output from a frequency analysis includes the following.

•

Vibrational frequencies.
•

Raman and IR activities and intensities (requires $rem DORAMAN).
•

Atomic masses.
•

Zero-point vibrational energy.
•

Translational, rotational, and vibrational, entropies and enthalpies.

Several other $rem variables are available that control the vibrational frequency analysis. In detail, they are:

DORAMAN

DORAMAN
       Controls calculation of Raman intensities. Requires JOBTYPE to be set to FREQ
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       FALSE Do not calculate Raman intensities. TRUE Do calculate Raman intensities.
RECOMMENDATION:
       None

VIBMAN_PRINT

VIBMAN_PRINT
       Controls level of extra print out for vibrational analysis.
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       1 Standard full information print out. If VCI is TRUE, overtones and combination bands are also printed. 3 Level 1 plus vibrational frequencies in atomic units. 4 Level 3 plus mass-weighted Hessian matrix, projected mass-weighted Hessian matrix. 6 Level 4 plus vectors for translations and rotations projection matrix.
RECOMMENDATION:
       Use the default.

CPSCF_NSEG

CPSCF_NSEG
       Controls the number of segments used to calculate the CPSCF equations.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Determine the number of segments based on the memory request and MEM_TOTAL $n$ User-defined. Use $n$ segments when solving the CPSCF equations.
RECOMMENDATION:
       Use the default.

Example 10.28 EDF1/6-31+G* optimization followed by vibrational analysis. Doing the vibrational analysis at a stationary point is necessary for the results to be valid.

$molecule
   0  1
   O
   C  1  co
   F  2  fc  1  fco
   H  2  hc  1  hco  3  180.0

   co  =   1.2
   fc  =   1.4
   hc  =   1.0
   fco = 120.0
   hco = 120.0
$end

$rem
   JOBTYPE    opt
   METHOD     edf1
   BASIS      6-31+G*
$end

@@@

$molecule
   read
$end

$rem
   JOBTYPE    freq
   METHOD     edf1
   BASIS      6-31+G*
$end

10.7.1.2 Interpreting the Output

Numerical values in the following discussion correspond to Example 10.7.1.1 and the referenced partition functions come from the textbook by McQuarrie & Simon. Note that Q-Chem assumes $T=298.15$ K and $P=1.00$ atm by default; see Section 10.7.2 for instructions on how to modify these choices.

The quantity listed as zero-point energy (ZPE),

Zero point vibrational energy: 12.695 kcal/mol

corresponds to $\text{ZPE}=\tfrac{1}{2}hc\nu N_{A}$ . Within Q-Chem the vibrational entropy ( $S_{\text{vib}}$ ) is computed from the bottom of the well rather than from the zero-point level, and thus includes the ZPE. This corresponds to a vibrational partition function

q_{\text{vib}}(T)=\prod_{K}\frac{e^{-\Theta_{\nu,K}/2T}}{1-e^{-\Theta_{\nu,K}/% T}}

(10.38)

where $K$ indexes vibrational modes and $\Theta_{K}=h\nu_{K}/k_{B}$ is the vibrational temperature of the $K$ th mode. This expression can be altered by a factor of $\Theta_{K}/2$ to start at the first vibrational level rather than the bottom of the well. The vibrational entropy is then

\displaystyle\begin{aligned} \displaystyle S_{\text{vib}}&\displaystyle=k_{B}T% \left(\frac{\partial\ln q_{\text{vib}}}{\partial T}\right)_{\!V}+k_{B}\ln q_{% \text{vib}}\\ &\displaystyle=k_{B}T\sum_{K}\left[\frac{\Theta_{K}/T}{e^{-\Theta_{K}/T}-1}-% \ln(1-e^{-\Theta_{K}/T})\right]\;,\end{aligned}

(10.39)

and for Example 10.7.1.1 it is reported as

Vibrational Entropy: 0.620 cal/mol.K

The vibrational enthalpy,

H_{\text{vib}}=c\sum_{K}\frac{\nu_{K}}{e^{\Theta_{K}/T}-1}\;,

(10.40)

is reported as

Vibrational Enthalpy: 12.839 kcal/mol

The rotational and translational enthalpy are multiples of $R T$ and then then

Total Enthalpy: 15.209 kcal/mol

is given by

H_{\text{total}}=H_{\text{vib}}+H_{\text{rot}}+H_{\text{trans}}+RT\;,

(10.41)

where the final contribution of $R T$ comes from the usual definition, $H=U+PV$ . Translational and rotational entropies can be computed from the corresponding ideal-gas partition functions; see Ref. .

For thermochemical calculations performed in implicit solvent (using models described in Section 11.2), there is a subtle point with some controversy attached. ⁵⁴⁷ Ho J., Klamt A., Coote M. L.
J. Phys. Chem. A
(2010), 114, pp. 13442. Link ^, ¹⁰⁹⁰ Ribeiro R. F. et al.
J. Phys. Chem. B
(2011), 115, pp. 14556. Link ^, ¹⁹⁵ Casasnovas R. et al.
Int. J. Quantum Chem.
(2014), 114, pp. 1350. Link ^, ⁵²⁹ Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519. Link This regards whether vibrational contributions to the free energy should be included or not. In nearly all cases, continuum solvation models are parameterized in order to reproduce experimental free energies of solvation ( $\Delta_{\text{solv}}G^{\circ}$ ) using rigid gas-phase geometries for the solute molecules. There is a potential double-counting problem if $-TS_{\text{vib}}$ from a vibrational frequency calculation in implicit solvent is added, which might be avoided by instead using gas-phase harmonic frequencies for the solute. ⁵⁴⁷ Ho J., Klamt A., Coote M. L.
J. Phys. Chem. A
(2010), 114, pp. 13442. Link However, the difference between these procedures is negligible ( $\sim 0.2$ kcal/mol). for the small-molecule data sets that are used to train implicit solvent models, ¹⁰⁹⁰ Ribeiro R. F. et al.
J. Phys. Chem. B
(2011), 115, pp. 14556. Link and only by using solution-phase vibrational frequencies can one obtain corrections to $S_{\text{vib}}$ arising from solvation-induced changes in geometry, which might be significant for larger molecules. Note that Q-Chem’s harmonic frequency engine assumes a gas-phase molecule (even in the presence of continuum solvent), so that rotational and translational contributions to the free energy are printed out in every case. These should not be included in solution-phase free energy differences.

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10.7.1 Overview

10.7.1.1 Job Control

10.7.1.2 Interpreting the Output