10.11 Vibrational Circular Dichroism (VCD)10.11 Vibrational Circular Dichroism (VCD)10.12 Finite-Field Calculation of (Hyper)Polarizabilities

10.11.1 Introduction

(April 13, 2024)

The VCD signals in dilute solution in terms of the differential molar extinction coefficient, $\Delta\epsilon=\epsilon_{\text{left}}-\epsilon_{\text{right}}$ , is given by

\Delta\epsilon=4\gamma\nu\sum_{i}\alpha_{g}R_{gi}f(\nu_{gi},\nu)\;,

(10.85)

where

\gamma=\frac{N_{A}\pi^{3}}{375\ln(10)hc}\;,

(10.86)

$N_{A}$ is Avogadro constant, $h$ is Planck’s constant and $c$ is the speed of light, $\nu$ is the frequency of the incident light, and $\alpha_{g}$ is the population of the ground state The quantity $R_{gi}$ is the rotational strength, with indices $g$ and $i$ represent the ground state and the $i$ th vibrational excited state, respectively; $f(\nu_{gi}$ is the lineshape function; and $\nu_{gi}$ is the transition frequency between vibrational states $g$ and $i$ . The rotational strength is given by

R_{gi}=\Im[\boldsymbol{P}_{i}\cdot\boldsymbol{M}_{i}]\;,

(10.87)

where $\Im[\cdots]$ means the the imaginary part of a complex number, $\boldsymbol{M}_{i}$ is the vibrational magnetic transition dipole vector, and $\boldsymbol{P}_{i}$ is the vibrational electric transition dipole vector. The latter can be expressed as

\boldsymbol{P}_{i}=\langle 0|\hat{\boldsymbol{\mu}}_{\text{e}}|1\rangle_{i}\\ =\sqrt{\frac{\hbar}{4\pi\nu_{i}}}\left.\left(\frac{\partial\langle\Psi_{g}|% \hat{\boldsymbol{\mu}}_{\text{e}}|\Psi_{g}\rangle}{\partial Q_{i}}\right)% \right|_{R_{0}}\;.

(10.88)

$\Psi_{g}$ denotes the ground state wave function, $Q_{i}$ is the $i$ -h normal mode, and this derivative is evaluated at the equilibrium geometry $R_{0}$ . The quantity

\hat{\bm{\mu}}_{\text{e}}=1+\sum_{\lambda}Z_{\lambda}\delta(r-R_{\lambda})er

(10.89)

is the electric transition dipole operator, where $Z_{\lambda}$ is the atomic number for atom $\lambda$ , located at position $R_{\lambda}$ .

Since the probability of the $1\leftarrow 0$ transition of the $i$ th normal mode is proportional to the modulus square of the corresponding electric transition dipole, $|\boldsymbol{P}_{i}|^{2}$ , the quantity $\boldsymbol{P}_{i}$ is evaluated in the frequency subroutine. Suppose the linear transform relation between the normal modes $Q_{i}$ and the atomic Cartesian displacement iis

X_{\lambda\alpha}=\sum_{i}S_{\lambda\alpha,i}\,Q_{i}

(10.90)

where $\lambda$ is the index for nuclei and $\alpha\in\{x,y,z\}$ . Matrix elements $S_{\lambda\alpha,i}$ define the normal modes and are obtained from a harmonic frequency calculation. We can define the atomic polar tensor (APT) having matrix elements

\boldsymbol{P}_{\lambda\alpha,\beta}=\left.\left(\frac{\partial\langle\Psi_{g}% |(\hat{\boldsymbol{\mu}}_{\text{e}})_{\beta}|\Psi_{g}\rangle}{\partial% \boldsymbol{X}_{\lambda\alpha}}\right)\right|_{R_{0}}\;.

(10.91)

Thus,

\tilde{\boldsymbol{P}}_{i\beta}=\sqrt{\frac{\hbar}{4\pi\nu_{i}}}\sum_{\lambda% \alpha}\boldsymbol{P}_{\lambda\alpha,\beta}\,\boldsymbol{S}_{\lambda\alpha,i}\;.

(10.92)

The quantity $\boldsymbol{P}_{\lambda\alpha,\beta}$ can be separated into the nuclear and electronic parts,

\boldsymbol{P}_{\lambda\alpha,\beta}=\boldsymbol{N}_{\lambda\alpha,\beta}+% \boldsymbol{E}_{\lambda\alpha,\beta}\;,

(10.93)

where

\boldsymbol{N}_{\lambda\alpha,\beta}=Z_{\lambda}e\delta_{\alpha\beta}

(10.94)

and

\boldsymbol{P}_{\lambda\alpha,\beta}=\left.\left(\frac{\partial\langle\Psi_{g}% |e\boldsymbol{r}_{\beta}|\Psi_{g}\rangle}{\partial\boldsymbol{X}_{\lambda% \alpha}}\right)\right|_{R_{0}}=2\left\langle\psi_{g}\bigg{|}e\boldsymbol{r}_{% \beta}\bigg{|}\frac{\partial\Psi_{g}}{\partial X_{\lambda\alpha}}\right\rangle\;.

(10.95)

Similarly, for the vibrational magnetic transition dipole $\tilde{\boldsymbol{M}}_{i}$ , we have

\tilde{\boldsymbol{M}}_{i\beta}=\langle 0|\hat{\boldsymbol{\mu}}_{\text{m}}|1% \rangle_{i\beta}\\ =-\sqrt{4\pi\hbar^{3}\nu_{i}}\sum_{\lambda\alpha}\boldsymbol{M}_{\lambda\alpha% ,\beta}\,S_{\lambda\alpha,i}

(10.96)

where

\hat{\boldsymbol{\mu}}_{\text{m}}=\frac{e}{2m}(\hat{r}\times\hat{p})

(10.97)

and $\boldsymbol{M}^{\lambda}_{\alpha\beta}$ is the atomic axial tensor (AAT). Moreover,

\boldsymbol{M}_{\lambda\alpha,\beta}=\boldsymbol{I}_{\lambda\alpha,\beta}+% \boldsymbol{J}_{\lambda\alpha,\beta}

(10.98)

where

\boldsymbol{I}_{\lambda\alpha,\beta}=\left\langle\left(\frac{\partial\Psi_{g}}% {\partial X_{\lambda\alpha}}\right)_{R_{0}}\bigg{|}\left(\frac{\partial\Psi_{g% }}{\partial H_{\beta}}\right)_{H_{\beta}=0}\right\rangle

(10.99)

and

\boldsymbol{J}_{\lambda\alpha,\beta}=\frac{i}{4\hbar c}\sum_{\gamma}\epsilon_{% \alpha\beta\gamma}Z_{\lambda}eR_{\lambda\gamma}\;.

(10.100)

The quantity $\epsilon_{\alpha\beta\gamma}$ is Levi-Civita symbol and $R_{\lambda\gamma}$ is some Cartesian component of $R_{\gamma}$ ( $\gamma\in\{x,y,z\}$ ). The derivative of the wavefunction with respect to the external magnetic field $H_{\beta}$ is evaluated with a perturbation $H^{\prime}=-\boldsymbol{\mu}_{m}\cdot H_{\beta}$ at the limit of $H_{\beta}\rightarrow 0$ . Note that $\tilde{M}_{i}$ is pure imaginary if real basis functions are used.

The working equation for rotational strength can be expressed as

R_{gi}=\hbar^{2}\Im\left[\sum_{\beta}\left(\sum_{\lambda\alpha}\boldsymbol{P}_% {\lambda\alpha,\beta}S_{\lambda\alpha,i}\cdot\\ \sum_{\lambda^{\prime}\alpha^{\prime}}\boldsymbol{M}_{\lambda^{\prime}\alpha^{% \prime},\beta^{\prime}}S_{\lambda^{\prime}\alpha^{\prime},i}\right)\right]

(10.101)

where $\lambda^{\prime}$ , $\alpha^{\prime}$ , and $\beta^{\prime}$ are dummy indices for atoms and Cartesian coordinates, respectively. The $\boldsymbol{I}^{\lambda}_{\alpha\beta}$ tensor can be rewritten as

\boldsymbol{I}_{\lambda\alpha,\beta}=\sum_{\mu\nu}\left(D_{\mu\nu}\Big{\langle% }\chi^{X_{\lambda\alpha}}_{\mu}\bigg{|}\chi^{H_{\beta}}_{\nu}\Big{\rangle}+D_{% \mu\nu}^{H_{\beta}}\Big{\langle}\chi^{X_{\lambda\alpha}}_{\mu}\bigg{|}\chi_{% \nu}\Big{\rangle}+D_{\mu\nu}^{X_{\lambda\alpha}}\Big{\langle}\chi_{\mu}\bigg{|% }\chi^{H_{\beta}}_{\nu}\Big{\rangle}+D_{\mu\nu}^{X_{\lambda\alpha},H_{\beta}}% \Big{\langle}\chi_{\mu}\bigg{|}\chi_{\nu}\Big{\rangle}\right)\;,

(10.102)

defining the intermediate quantities


$\displaystyle D_{\mu\nu}$	$\displaystyle=\sum_{i}^{\text{occ}}c^{\ast}_{\mu i}c_{\nu i}$	(10.103a)
$\displaystyle D_{\mu\nu}^{H_{\beta}}$	$\displaystyle=\sum_{i}^{\text{occ}}c^{\ast}_{\mu i}c_{\nu i}^{H_{\beta}}$	(10.103b)
$\displaystyle D_{\mu\nu}^{X_{\lambda\alpha}}$	$\displaystyle=\sum_{i}^{\text{occ}}c^{X_{\lambda\alpha}\ast}_{\mu i}c_{\nu i}$	(10.103c)
$\displaystyle D_{\mu\nu}^{X_{\lambda\alpha},H_{\beta}}$	$\displaystyle=\sum_{i}c^{X_{\lambda\alpha}\ast}_{\mu i}c_{\nu i}^{H_{\beta}}$	(10.103d)

and


$\displaystyle c_{\nu i}^{H_{\beta}}$	$\displaystyle=\frac{\partial c_{\nu i}}{\partial{H_{\beta}}}\bigg{\|}_{H_{\beta% }=0}$	(10.104a)
$\displaystyle c^{X_{\lambda\alpha}}_{\mu i}$	$\displaystyle=\frac{\partial c_{\mu i}}{\partial{X_{\lambda\alpha}}}\bigg{\|}_{% R_{0}=0}$	(10.104b)

where $\chi_{\mu,\nu}$ are AO basis functions and $c_{\mu,\nu i}$ are MO coefficients. Superscripts of $c$ indicate MO coefficient derivatives with respect to nuclear position and magnetic field. Derivatives of the MO coefficient are obtained by solving CPHF/CPKS equations.

Electric transition dipoles are origin-independent while magnetic transition dipoles are origin-dependent with finite basis sets. In order to obtain origin-independent (gauge-invariant) VCD properties, one can employ the explicit field-dependent GIAO basis functions:

\chi_{\mu,\text{GIAO}}(\boldsymbol{H})=\text{exp}\left[-\frac{i}{2c}(% \boldsymbol{H}\times\boldsymbol{R}_{\mu})\cdot\boldsymbol{r}\right]\chi_{\mu}

(10.105)

To extract the VCD properties, one should (at a minimum) carry out a frequency analysis with the system. Several available $rem variables include:

VCD

VCD
       Controls calculation of the VCD signals. Requires JOBTYPE to be set to FREQ
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       FALSE Do not calculate the VCD properties. TRUE Do calculate the VCD properties.
RECOMMENDATION:
       None

VCD_PRINT

VCD_PRINT
       Controls level of extra print out for the VCD calculations.
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       1 Standard full information print out. 2 Electronic part of AAT.
RECOMMENDATION:
       Use the default.

Example 10.49 An PBE/STO-3G optimization, followed by a VCD calculation.

$comment
(-)-camphore
$end

$molecule
0 1
 O  -2.5217    0.3747   -0.2628
 C   0.9690    0.2807   -0.4137
 C  -0.2122    0.5268    0.5653
 C   0.5981   -1.2183   -0.5902
 C  -0.0177   -0.5488    1.6588
 C   0.5627   -1.7425    0.8680
 C  -0.8632   -1.1910   -1.1027
 C  -1.3692   -0.0289   -0.2712
 C   0.9102    1.0940   -1.7214
 C   2.3671    0.5229    0.1910
 C  -0.4232    1.9305    1.0788
 H   1.2655   -1.7945   -1.2356
 H  -0.9615   -0.8246    2.1436
 H   0.6774   -0.2257    2.4410
 H   1.5668   -1.9925    1.2263
 H  -0.0563   -2.6395    0.9757
 H  -1.4099   -2.1082   -0.8724
 H  -0.9272   -0.9564   -2.1673
 H   1.1849    2.1396   -1.5401
 H  -0.0766    1.1153   -2.1913
 H   1.6143    0.6876   -2.4563
 H   2.5320    0.0303    1.1519
 H   2.5399    1.5928    0.3537
 H   3.1441    0.1597   -0.4916
 H   0.4499    2.2884    1.6335
 H  -0.6167    2.6304    0.2592
 H  -1.2875    1.9748    1.7507
$end

$rem
   JOBTYPE opt
   BASIS sto-3g
   METHOD pbe
point_group_symmetry = False
integral_symmetry = false
   no_reorient true
$end

@@@

$molecule
read
$end

$rem
   JOBTYPE = freq
   BASIS  = sto-3g
   METHOD  =  PBE
point_group_symmetry = False
integral_symmetry = false
   VCD = 1
   VCD_PRINT = 2
   no_reorient true
$end

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10.11.1 Introduction