10.7 Harmonic Vibrational Analysis 10.7.1 Introduction 10.7.3 Isotopic Substitutions

10.7.2 Treatment of Low-Frequency Vibrational Modes

(April 13, 2024)

Low-frequency vibrational modes usually emerge due to hindered or near-free rotations around a single bond within a molecule. The harmonic approximation is problematic for such internal rotations and yields an infinite vibrational entropy $S_{\mathrm{vib}}$ in the limit of vanishing frequencies. To fix this issue, Grimme ⁴³⁹ Grimme S.
Chem. Eur. J
(2012), 18, pp. 9955. Link proposed to enforce a finite vibrational entropy by interpolating between the entropy of the free rotor $S_{\mathrm{FR}}$ and the harmonic vibrational entropy $S_{\mathrm{HO}}$ as

S_{\mathrm{vib}}(\nu_{i})=\big{(}1-\omega(\nu_{i})\big{)}S_{\mathrm{FR}}(\nu_{% i})+\omega(\nu_{i})S_{\mathrm{HO}}(\nu_{i}),

(10.25)

where

\omega(\nu_{i})=\frac{1}{1+(\nu_{0}/\nu_{i})^{\alpha}}

(10.26)

is the Chai-Head-Gordon damping function with $\alpha$ and $\nu_{0}$ as fixed parameters. ¹⁹⁵ Chai J.-D., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2008), 10, pp. 6615. Link It is used as a weighting function and allows for a smooth transition from the free rotor entropy at small frequencies to the harmonic vibrational entropy for frequencies $\nu_{i}$ above the cutoff $\nu_{0}$ . Head-Gordon and coworkers ⁷⁴³ Li Y.-P. et al.
J. Phys. Chem. C
(2015), 119, pp. 1840. Link later extended this by also interpolating the vibrational enthalpic contributions from a free rotor $H_{\mathrm{FR}}$ to that of a harmonic oscillator $H_{\mathrm{HO}}$ with zero-point vibrational energy $H_{\mathrm{ZPVE}}$ as

H_{\mathrm{vib}}(\nu_{i})=\big{(}1-\omega(\nu_{i})\big{)}H_{\mathrm{FR}}(\nu_{% i})+\omega(\nu_{i})\big{(}H_{\mathrm{HO}}(\nu_{i})+H_{\mathrm{ZPVE}}(\nu_{i})% \big{)}\,

(10.27)

in the same manner. This again reduces the error associated with treating translational and rotational degrees of freedom as low-frequency vibrations, which is especially important for the adsorption or association of larger molecules. This procedure is known as the quasi-rigid-rotor-harmonic-oscillator (qRRHO) approach.

The qRRHO scheme is the default in Q-Chem, and all thermodynamic quantities are printed for the RRHO (without interpolations) and qRRHO schemes at standard temperature and pressure (298.15 K and 1.00 atm). To change the latter, the user can specify an $isotopes section (see examples and Section 10.7.3 for details). $\alpha$ and $\omega_{0}$ for both interpolator functions can be modified through QRRHO_ALPHA and QRRHO_OMEGA_CUTOFF respectively.

QRRHO_ALPHA

QRRHO_ALPHA
       Specifies the exponent in the Chai-Head-Gordon damping function used for interpolating the vibrational enthalpy and entropy in the qRRHO scheme. Specify MRRHO_ALPHA to change the exponent for the entropy interpolation separately.
TYPE:
       INTEGER
DEFAULT:
       4
OPTIONS:
       $\alpha$ Dimensionless interpolator exponent used in the qRRHO scheme.
RECOMMENDATION:
       Use the default.

QRRHO_OMEGA_CUTOFF

QRRHO_OMEGA_CUTOFF
       Sets the frequency cutoff in the Chai-Head-Gordon damping function for interpolating the vibrational enthalpy and entropy in the qRRHO scheme. Specify MRRHO_OMEGA_CUTOFF to change the frequency cutoff for the entropy interpolation separately.
TYPE:
       INTEGER
DEFAULT:
       100
OPTIONS:
       $\omega_{0}$ Interpolator cutoff frequency used in the qRRHO scheme in $\mathrm{cm}^{-1}$ .
RECOMMENDATION:
       Use the default.

Example 10.26 Harmonic vibrational analysis at the HF/3-21G level of theory, where the thermodynamic properties for RRHO and qRRHO are printed at 298.15 K and 1.00 atm.

$molecule
0 1
C     1.682185104800      0.240320237000      0.000000000000
O     0.894276382600      1.089998584000      0.000000000000
O     2.480949717100     -0.590833069200      0.000000000000
C    -1.682185094800     -0.240320333400      0.000000000000
O    -0.894276273900     -1.089998812500      0.000000000000
O    -2.480949835800      0.590833394100      0.000000000000
$end

$rem
JOBTYPE         opt
METHOD          hf
BASIS           3-21G
integral_symmetry false
point_group_symmetry False
$end

@@@

$molecule
read
$end

$rem
JOBTYPE         freq
METHOD          hf
BASIS           3-21G
integral_symmetry false
point_group_symmetry False
$end

Example 10.27 Harmonic vibrational analysis at the HF/3-21G level of theory, where the thermodynamic properties for RRHO and qRRHO are printed at standard temperature and pressure as well as two additional temperatures (273.15 K and 313.15 K). Please see Section 10.7.3 for further details on the $isotopes section.

$molecule
0 1
C     1.682185104800      0.240320237000      0.000000000000
O     0.894276382600      1.089998584000      0.000000000000
O     2.480949717100     -0.590833069200      0.000000000000
C    -1.682185094800     -0.240320333400      0.000000000000
O    -0.894276273900     -1.089998812500      0.000000000000
O    -2.480949835800      0.590833394100      0.000000000000
$end

$rem
JOBTYPE         opt
METHOD          hf
BASIS           3-21G
integral_symmetry false
point_group_symmetry False
$end

@@@

$molecule
read
$end

$rem
JOBTYPE         freq
METHOD          hf
BASIS           3-21G
integral_symmetry false
point_group_symmetry False
ISOTOPES        true
$end

$isotopes
2 1
0 273.15 1.0
0 313.15 1.0
$end

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10.7.2 Treatment of Low-Frequency Vibrational Modes