VCD Analysis: VCDtools
**********************

The auxiliary program *VCDtools* provides insight regarding the origin
of the VCD intensity of a given normal mode. This is done by analyzing
and visualizing the various contributions to the total Electric (E)
and Magnetic (M) Dipole Transition Moments (DTM). *e.g.*, the
contributions from atoms, electrons, nuclei and molecular fragments.
The *VCDtools* program is accessible through the AMS-GUI and allows
one to:

1. Scale the size of the atoms with the magnitude of their atomic EDTM
   or MDTM contributions.

2. Visualize the various contributions to the EDTMs and MDTMs by
   arrows.

3. Perform a General Coupled Oscillator (GCO) analysis to evaluate the
   interactions between the different parts of a molecule.

4. Decompose the total EDTM and MDTM into nuclear and electronic
   contributions to gain insight into effects induced in the VCD
   spectra by charge transfer.

5. Perform a VCD robustness analysis.

See also:

  Tutorial: Analysis of the VCD spectrum of Oxirane with VCDtools

*VCDtools* requires only the standard result file adf.rkf from a VCD
calculation using the *ADF* engine. Once this file is opened with
*AMSspectra* the *VCDtools* program will be accessible through the
standard user interface of *AMSspectra*.

*VCDtools* produces one ASCII file, VCDtools.out where all output from
the program is printed. This file will be located in the *.results*
folder in the directory of the opened adf.rkf result file.
Additionally, VCDtools will run in the background every time a new
normal mode is selected. It then computes the atomic EDTM and MDTM
which can be displayed inside *AMSspectra* in various ways.

The following references should be cited when publishing results
obtained with VCDtools Refs. [1] [2] [3].


General Theory
==============

The VCD intensity associated with the fundamental vibrational
transition for a given normal mode j  is given by the rotational
strength (RS):

   R_{01}(j) = -i \vec{E}^{tot}_{01}(j) \cdot \vec{M}^{tot}_{10}(j)

where E^{tot}_{01}(j) and M^{tot}_{10}(j) are the total EDTM and MDTM
of normal mode j , i  is the unit imaginary number.

Within the harmonic approximation the total EDTM and MDTM can be
written as sums of atomic contributions:

   \vec{E}^{tot}_{01}(j) = \sum_{\lambda = 1}^N
   \vec{E}^{\lambda}_{01}(j) \\ \vec{M}^{tot}_{10}(j) = \sum_{\lambda
   = 1}^N \vec{M}^{\lambda}_{10}(j)

where N is the total number of atoms in the molecule and \lambda runs
over all atoms.


General Coupled Oscillator Analysis
===================================

The GCO analysis computes the contribution to total rotational
strength from the interaction of two molecular fragments. This
information provides important insight regarding the source of the VCD
intensity and the robustness of the computed VCD sign. In the
following, a brief summary of the GCO theory originally published in
Ref. [1] is given.

Following the original coupled oscillator VCD mechanism, the atoms of
a molecule are grouped into three fragments: **A**, **B**, and **R**.
The fragments **A** and **B** represent the important Coupling
Oscillator (CO) fragments, while fragment **R** is the ancillary
fragment containing the rest of the atoms. As a result, the total RS
can be decomposed in three components:

   R_{01}(j)  =  R_{01}^{\mathbf{GCO}}(j) + R_{01}^{\mathbf{IF}}(j) +
   R_{01}^{\mathbf{R}}(j)

The first contribution R_{01}^{\mathbf{GCO}}(j), is the GCO
contribution to the RS and is given by the interaction between the
EDTM and MDTM of fragments **A** and **B**:

   R_{01}^{\mathbf{GCO}}(j)  =  - i \cdot \left [
   \vec{E}^{\mathbf{A}}_{01}(j) \cdot \vec{M}^{\mathbf{B}}_{10}(j)  +
   \vec{E}^{\mathbf{B}}_{01}(j) \cdot \vec{M}^{\mathbf{A}}_{10}(j)
   \right]

The R_{01}^{\mathbf{IF}}(j) term represents the contribution from the
individual fragments (IF) **A** and **B** without any interaction with
other fragments:

   R_{01}^{\mathbf{IF}}(j)   = - i \cdot \left [
   \vec{E}^{\mathbf{A}}_{01}(j) \cdot \vec{M}^{\mathbf{A}}_{10}(j)  +
   \vec{E}^{\mathbf{B}}_{01}(j) \cdot \vec{M}^{\mathbf{B}}_{10}(j)
   \right]

and the R_{01}^{\mathbf{R}}(j) contribution contains all contributions
from fragment **R**:

   R_{01}^{\mathbf{R}}(j)   = & - i \cdot \left[
   \vec{E}^{\mathbf{A}}_{01}(j) \cdot \vec{M}^{\mathbf{R}}_{10}(j)  +
   \vec{E}^{\mathbf{R}}_{01}(j) \cdot \vec{M}^{\mathbf{A}}_{10}(j)
   \right] \\            & - i \cdot \left[
   \vec{E}^{\mathbf{B}}_{01}(j) \cdot \vec{M}^{\mathbf{R}}_{10}(j)  +
   \vec{E}^{\mathbf{R}}_{01}(j) \cdot \vec{M}^{\mathbf{B}}_{10}(j) +
   \vec{E}^{\mathbf{R}}_{01}(j) \cdot \vec{M}^{\mathbf{R}}_{10}(j)
   \right]

In the above equations \vec{E}^{\mathbf{X}}_{01}(j) and
\vec{M}^{\mathbf{X}}_{01}(j), with \mathbf{X}=(\mathbf{A}, \mathbf{B},
\mathbf{R}), are the EDTM and MDTM associated with the three
fragments.

Using the origin dependency of the MDTM, the GCO contribution can be
rewritten in a form similar to the original CO term:

   R_{01}^{\mathbf{GCO}}(j)  = \frac{\pi \nu_j}{c} \cdot
   \vec{Y}^{\mathbf{GCO} }(j) \cdot \left[ \vec{E}^\mathbf{A}_{01}(j)
   \times \vec{E}^\mathbf{B}_{01}(j) \right]

where \vec{Y}^{\mathbf{GCO} }(j) is the general coupled oscillator
vector.

As shown in ref. [1] most normal modes of a molecule (i.e. not just
the carbonyl stretching modes) can be interpreted in terms of the GCO
mechanism. That is, for most normal modes one can define the GCO
fragments **A** and **B** fragments in such a way that the
R_{01}^{\mathbf{GCO}}(j) term represent the dominant contributions to
the rotational strengths. This second expression for
R_{01}^{\mathbf{GCO}}(j) does not depend on the MDTM (which is origin
dependent). As such, it allows one to interpret the VCD intensity of a
mode in terms of interacting EDTMs that are associated with the
various moieties of a molecule and their relative orientation.

The identification of the GCO fragments (which are normal mode
dependent) is not always trivial, especially, in molecules without
symmetry. Consequently, *VCDtools* offers several options for dividing
the molecule into fragments. In *AMSspectra* these fragments are
referred to as *'Regions'* in analogy to the regions in the *AMSinput*
program. Under the **Regions**-menu there are many options to set,
alter and save the fragments. Additionally, *VCDtools* is able to make
a guess for fragments **A** and **B** as will be discussed in more
detail below.


Available options
=================

Besides the standard calculation and visualization of the atomic
contributions to the normal mode (NM) motion, EDTM and MDTM,
*VCDtools* offers three more advanced tools that can be utilized when
analyzing VCD spectra inside *AMSspectra:*

* GCO Analysis

* Guess GCO Fragments

* NM Localization on Regions

The first option, *"GCO Analysis"*, uses the above equations to
decompose the RS in its different contributions. Printing both the
values and important angles between the vectors. Before this option
can be run two regions inside the molecule should be defined as
fragments **A** and **B**. It is important that these regions do not
contain the same atoms. Also since the decomposition is different for
each NM, a specific NM should be selected.

The second option, *"Guess Fragments"*, guesses which atoms belong to
the fragments **A** and **B** for a selected normal mode. In doing so
the R_{01}^{\mathbf{GCO}}(j) is maximized while keeping
R_{01}^{\mathbf{R}}(j) low. Additionally, it ensures that the
fragments are localized on a part of the molecule.

The third option, *"NM Localization on Regions"*, computes the
percentage of the mass-weighted normal mode motion that is located on
the atoms in the selected regions. One or multiple regions can be
computed at the same time and the localizations are determined for all
modes within a selected frequency window.

[1] V.P. Nicu, *Revisiting an old concept: the coupled oscillator
    model for VCD. Part 1: the generalised coupled oscillator
    mechanism and its intrinsic connection to the strength of VCD
    signals*, Physical Chemistry Chemical Physics 18, 21202 (2016)

[2] V.P. Nicu, J. Neugebauer and E.J. Baerends, *Effects of Complex
    Formation on Vibrational Circular Dichroism Spectra*, Journal of
    Physical Chemistry A 112, 6978 (2008)

[3] M.A.J. Koenis, O. Visser, L. Visscher, W.J. Buma, V.P. Nicu, *GUI
    Implementation of VCDtools, A Program to Analyze Computed
    Vibrational Circular Dichroism Spectra*, J. Chem. Inf. Model 60,
    259 (2020)
