Subject: Anisotropic Hyperfine Summary

From: [email-masked] (Dr Fiona Bettens)
Date: Fri, 9 May 1997 20:16:39 +1000

Dear Netters,

My original query was:

>Is anyone aware of any software that will either take the
>molecular-orbital expansion coefficients produced in an ab initio
>calculation or do the ab initio calculation itself to calculate the
>expectation values of the anisotropic hyperfine tensor.  By this I
>mean the tensor that represents the coupling of the electronic
>mangnetic dipole with the nuclear magnetic dipole.  I am interested in
>transition metal complexes with more than one unpaired electron, this
>complicates matters somewhat.  I suspect MELDF does what I want, but
>it is difficult to find documentation on exactly how it does it.

*******************
*******************

The anisotropic hfs is proportional to the efg at the nucleus produced
by a hypothetical charge density representing the spin density.
Therefore in G94 the input card

#P cisd/6-31g** scf=direct density=all IOP(6/26=4,6/17=2) prop=efg

is sufficient to generate the anisotropic hf tensor for all nuclei,
subject to the appropiate proportionality factors (available from a
standard textbook).

Sincerely

Rod Macrae

*******************
*******************

First note that in my reply I am \underline{not} talking about the
nuclear quadrupole splitting, which is evaluated by calculating the
true efg at the nucleus (i.e.  not a spin property) and multiplying it
by a nuclear term extracted from a spin hamiltonian treatment of the
nuclear component of the interaction.  However, this does give a guide
as to how to think about the nucleus-electron dipole-dipole hyperfine
interaction.  The expression will contain an operator (describing the
angular and radial properties of the interaction) evaluated at the
nucleus.  This will be multiplied by a factor in which the nuclear
part of the interaction, expressed in terms of a spin hamiltonian,
will be embedded.

The efg operator takes the form $(3 \cos^{2}\theta_{J}-1)/r^{3}$.  (I
will write mathematical expressions in Latex form so that you can
print them out and look at them if you want.)  The nuclear quadrupole
coupling has electronic component $V_{zz} = {\langle} \psi_{e} | {\cal
V}_{zz} | \psi_{e} {\rangle}$.

In the case of the dipolar hyperfine interaction the angular
expression in terms of the nuclear-electronic spin hamiltonian is
given by (Atkins 14.11.2):

\begin{displaymath}
\hat{\cal H}_{hf} = \mu_{0} g_{e} \gamma_{e} \gamma_{N} \frac{({\bf
s}\cdot{\bf I} - 3 {\bf s}\cdot \hat{\bf r} \hat{\bf r} \cdot {\bf
I})}{r^{3}}.
\end{displaymath}

At this point what generally happens is that a \underline{high field
approximation} is made (aligned spins), and all spin components other
than the z components disappear.  What this explicitly means is that
the Zeeman interaction is much larger than the hyperfine interaction,
and only the largest terms in the expansion of the dipole-dipole
interaction - see e.g.  Abragam p.  104 - need be considered (as a
perturbation of the Zeeman interaction).  It is this procedure that
leads to a dipole-dipole coupling proportional to $\frac{1}{r^{3}} (1
- 3 \cos^{2}\theta ) I_Z s_Z$, and therefore essentially proportional
to an "electric field gradient" constructed from the spin density
(rather than the charge density) distribution.  This is the conceptual
procedure behind my previous answer.

I don't know offhand if 1) this approximation procedure is less valid
for I > 1/2, or 2) there is a commercially-available program which
performs the calculation using the full expression.  (If you find one,
I'd very much appreciate it if you'd tell me.)

Sincerely

Rod

Sources: Atkins, Quantum Mechanics,
Lucken, Nuclear Quadrupolar Interactions, Abragam, Principles of Nuclear
Magnetism.

R. M. Macrae,
Muon Science Laboratory
Institute of Physical and Chemical Research (RIKEN)
e-mail: [email-masked] (normal)
 and  : [email-masked] (MIME-encoded)
 Tel  : (81) 484 62 1111 ext 3336
 Fax  : (81) 484 62 4648

*******************
*******************

The properties package in MELDF deals with some basic expectaction
values, e.g. del(A)s, the del operator on center A evaluated over the
spin density. For the anisotropic components of the hyperfine
interaction it needs operators like xx/r**5, xy/r**5, yy/r**5, etc.
also evaluated over the spin operator. The latter values are combined
to form the anisotropic hyperfine (3x3) matrix in the molecular axis
system, using a convention that requires the trace of this matrix to
have a zero trace.

Some of the components are defined as follows:
A(xx) =  (3/2)*(x2-y2)/r5 - (0.5)*(z2-r2)/r5
A(yy) = -(3/2)*(x2-yy)/r5 - (0.5)*(z2-r2)/r5
A(zz) = (z2-r2)/r5

Ernest Davidson and I have written a chapter in a book on the general
topic of computing hyperfine parameters. It's in Theoretical Models
of Chemical Bonding, Part 3, Spinger Verlag, 1991.

David Feller |
Environmental Molecular Sciences Laboratory | Battelle Pacific
Northwest National Lab |
Mail Stop K1-90 | e-mail:[email-masked]
906 Battelle Blvd | Fax: (509)-375-6631
Richland, WA 99352 |

*******************
*******************

The folks at Gaussian should have responded by now, but they haven't,
and I've given up on them. But I think that G94 really does calculate
the EFG tensor properly. IOp(6/17=2) selects the spin density, rather
than the charge density, and IOp(6/26=4) causes only the electronic
component of the integrals to be calculated, which is what we want. I
have used the EFG to calculate the anisotropic coupling for the
methyl radical as was done by Adamo, Barone, and Fortunelli in J.
Chem. Phys. 102 (1995) 384, and I can reproduce their values almost
exactly ( I think they used a slightly different B3LYP functional ).
I then calculated the couplings for an organometallic ion that I've
been working on, and I get values that are within 4-18% of experiment
(which, incidentally, is the same margin of error as reported by
Adamo, et al. for the methyl radical). This looks pretty good, but
I'm trying to improve the results.

The formula I used is:

       g B g_n B_n dE
T_ii = ----------- --
       a^3 h 10^13 dq_i

where g and g_n are the electron and nuclear g factors, B and B_n are
the corresponding magnetons, a is the bohr radius, and since the
tensor is already diagonalized, only the _ii elements are there. G94
reports dE/dq_i in atomic units, which means (1/bohr^3), and I've
converted this to MHz for T_ii with the terms in the denominator
(there is a factor of 10^7 in the denominator, too, from the
permeability constant).

Dale Braden
Department of Chemistry
University of Oregon
Eugene, OR 97403-1253
[email-masked]

*******************
*******************

Here is Dan Chipman's response to my query about the problem with the
x and y components of the spin operators, spin contamination, etc.

Dale Braden

---------- Forwarded message ----------

Dale,

It does indeed appear that G94 will do what you want by setting the
options you specify. Even so, I would suggest that you make a check
by trying to reproduce some number in the literature. I have found
that the G94 manual is not always in concordance what what actually
happens in the code.

The x,y,z indices on the S_x, S_y, and S_z spin operators refer to
the components relative to the applied magnetic field, which are
generally unrelated to the X,Y,Z coordinates of atoms in the
molecule-fixed frame. Usually, S_x and S_y only produce small
second-order effects in the hyperfine interaction. For the normal
first-order effects, one only need consider S_z. For a good
discussion of this, look at the old but good textbook "Introduction
to Magnetic Resonance" by Carrington and MacLachlan.

"Averaging over spin density" is a somewhat different concept than
what you indicate. It means taking the difference between values
obtained from the sum of all alpha-spin orbitals and the sum of all
beta-spin orbitals. Averaging over the total density, on the other
hand, means adding the alpha and beta contributions instead of
subtracting them.

Unfortunately, taking account of spin contamination is a difficult
problem. There is no simple "renormalization" correction that would
take care of it. Some people suggest projecting the incorrect spin
terms out of the wave function, but that is not really satisfactory
either. For isotropic hyperfine interactions, neither
unrestricted-Hartree Fock nor any of its spin-projected variants is
reliable. However, anisotropic hyperfine interactions are often
dominated by the singly-occupied molecular orbital in which case UHF,
PUHF, etc. may all work fairly well. That is, when the contribution
of the SOMO is dominant, one need not worry about spin contamination
effects.

If spin contamination is a problem, then one can either resort to
spin-restricted approaches, as I usually do, or to correlated methods
based on UHF such as UQCISD adn UQCISD(T) that automatically remove
most of the spin contamination. Unfortunately, these latter methods
are quite expensive.

Dan

*******************
*******************

I have received several requests about the computation of ESR
anisotropic coupling constants by Gaussian94. Since this seems a
general question I directly post this message for the whole CCL
community. These constants are nothing else than field gradients
computed with the spin rather than the total electronic density and
not including the nuclear contribution. Furthermore the resulting
tensor must be put in the zero-trace form. In gaussian94 it is
possible to force these options by setting in the keyword list the
following items:
PROP IOP(6/17=2,6/26=4)

Here follows an input and the relevant part of the output for a
STO-3G computation of H2NO

----------------------------------------------------------------------

#UHF/PROP IOP(6/17=2) IOP(6/26=4)

H2NO

0 2
X
X 1 1.0
N 2 1.0 1 90.0
H 3 NH  2 90.0  1  THETA
H 3 NH  2 90.0  1 -THETA
O 3 NO  2 ALPHA 1  180.0

NH=1.0179
NO=1.2778
THETA=58.9235
ALPHA=110.0

----------------------------------------------------------------------

Fermi contact analysis (atomic units).
1
1 N  .042923
2 H -.005038
3 H -.005038
4 O  .098025

**********************************************************************

Electrostatic Properties Using The SCF Density

**********************************************************************

Warning! Using spin rather than total density!

--- Only the electronic contributions will be computed ---

-----------------------------------------------------
Center ---- Electric Field Gradient ----
           XX        YY        ZZ
-----------------------------------------------------
1 Atom  .119043  -.295145  -.363285
2 Atom  .002506   .031422   .029384
3 Atom  .002506   .031422   .029384
4 Atom 2.130337 -1.663300 -1.698867
-----------------------------------------------------
-----------------------------------------------------
Center ---- Electric Field Gradient ----
           ( tensor representation )
          3XX-RR    3YY-RR    3ZZ-RR
-----------------------------------------------------
1 Atom   .298838  -.115349  -.183489
2 Atom  -.018598   .010318   .008280
3 Atom  -.018598   .010318   .008280
4 Atom  2.540947 -1.252690 -1.288257
-----------------------------------------------------

these are the principal values of anisotropic coupling constants

----------------------------------------------------------------------

Since the directions of principal moments are often significant and
transforma- tion to more conventional units can be performed once for
ever, I have modified the links 601 and 602 of gaussian to obtain the
following output for the same input

----------------------------------------------------------------------
----------------------------------------------------------------------

Isotropic Fermi Contact Couplings

----------------------------------------------------------------------

  Atom     a.u.  MegaHertz   Gauss    10(-4) cm-1

1 N(14)  .04292  13.86856   4.94865   4.62605
2 H     -.00504 -22.51979  -8.03562  -7.51179
3 H     -.00504 -22.51979  -8.03562  -7.51179
4 O(17)  .09802 -59.42481 -21.20426 -19.82198

----------------------------------------------------------------------

**********************************************************************

Electrostatic Properties Using The SCF Density

**********************************************************************

Warning! Using spin rather than total density!

--- Only the electronic contributions will be computed ---

Atomic Center 1 is at -.021142   .543231  .000000
Atomic Center 2 is at  .158563  1.036966  .871810
Atomic Center 3 is at  .158563  1.036966 -.871810
Atomic Center 4 is at -.021142  -.734569  .000000

-----------------------------------------------------
Center ---- Spin Dipole Couplings ----
          3XX-RR    3YY-RR    3ZZ-RR
-----------------------------------------------------
1 Atom   .298838  -.115349  -.183489
2 Atom  -.018598   .010318   .008280
3 Atom  -.018598   .010318   .008280
4 Atom  2.540947 -1.252690 -1.288257
-----------------------------------------------------
Center ---- Spin Dipole Couplings ----
            XY        XZ        YZ
-----------------------------------------------------
1 Atom  -.074772   .000000   .000000
2 Atom   .004800   .007098   .035617
3 Atom   .004800  -.007098  -.035617
4 Atom  -.099769   .000000   .000000

----------------------------------------------------------------------
     Anisotropic Spin Dipole Couplings in Principal Axis System
----------------------------------------------------------------------

Atom             a.u.  MegaHertz Gauss   10(-4) cm-1        Axes

        Baa    -.1835   -7.077  -2.525  -2.361      .0000  .0000 1.0000
1 N(14) Bbb    -.1284   -4.953  -1.767  -1.652      .1724  .9850  .0000
        Bcc     .3119   12.030   4.293   4.013      .9850 -.1724  .0000
       1/R**3  -.5394  -20.803  -7.423  -6.939

        Baa    -.0268  -14.276  -5.094  -4.762     -.2361 -.6571  .7158
2 H     Bbb    -.0193  -10.278  -3.667  -3.428      .9631 -.2560  .0828
        Bcc     .0460   24.554   8.762   8.190      .1288  .7090  .6933
       1/R**3   .0633   33.780  12.053  11.268

        Baa    -.0268  -14.276  -5.094  -4.762      .2361  .6571  .7158
3 H     Bbb    -.0193  -10.278  -3.667  -3.428      .9631 -.2560 -.0828
        Bcc     .0460   24.554   8.762   8.190      .1288  .7090 -.6933
       1/R**3   .0633   33.780  12.053  11.268

        Baa   -1.2883   93.221  33.264  31.095      .0000  .0000 1.0000
4 O(17) Bbb   -1.2553   90.837  32.413  30.300      .0263  .9997  .0000
        Bcc    2.5436 -184.059 -65.677 -61.395      .9997 -.0263  .0000
       1/R**3 -1.2318   89.138  31.807  29.733

----------------------------------------------------------------------

----------------------------------------------------------------------

Vincenzo Barone |
Professor of Theoretical Chemistry |
Dipartimento di Chimica | tel. +39-81-5476503
Universita' Federico II | fax +39-81-5527771
via Mezzocannone 4 | e-mail [email-masked]
I-80134 Napoli |
Italy   |
______________________________________________________________________

Below is a LaTeX summary of my findings to the original question:
****************************************************
\documentstyle{article}

\begin{document}

\title{Calculation of the Anisotropic Dipolar Coupling Constants}
\author{Fiona L. Bettens and Ryan P. A. Bettens}
\date{26th March 1997}
\maketitle

\section{Macroscopic Hamiltonian}

The anisotropic hyperfine coupling constants, $A_{\tau\upsilon}$, are
phenomenological constants utilized in a ``so called'' spin Hamiltonian, below
(see [\ref{bib:car67}] for example).  This Hamiltonian is written for one
nucleus of nonzero spin, $I_{\upsilon}$,

\begin{equation}
\hat{\cal H}_{\rm macro} = \mbox{\bf I}^{\dagger} \mbox{\bf A}
  \mbox{\bf S} =
  \sum_{\tau\upsilon} A_{\tau\upsilon} \hat{I}_{\tau} \hat{S}_{\upsilon}
\end{equation}

\noindent where the $\tau$ and $\upsilon$ represent the spaced fixed $X$, $Y$
and $Z$ axes and the operators $\hat{S}_{\upsilon}$ and $\hat{I}_{\tau}$
represent the components of {\it total} electronic spin and nuclear spin
respectively.  The spin Hamiltonian matrix is set up using the product basis,

\begin{equation}
|S, M_{S}; I, M_{I}\rangle = |S, M_{S}\rangle |I, M_{I}\rangle .
\label{macro2}
\end{equation}

\section{Microscopic Hamiltonian}

The Hamiltonian describing the {\it classical} interaction energy between a
nuclear magnetic dipole and the magnetic dipoles of $n$ electrons is
[\ref{bib:bev71}],

\begin{equation}
\hat{\cal H}_{\rm micro} = -\frac{\mu_{0}}{4\pi}g\beta g_{N}\beta_{N}
  \sum_{i=1}^{n}
  \left\{ \frac{{\bf I}\cdot{\bf s_{i}}}{{r_{i}}^{3}} -
    \frac{3({\bf I}\cdot{\bf r_{i}})({\bf s_{i}}\cdot{\bf r_{i}})}
    {{r_{i}}^{5}}
  \right\}
\label{micro1}
\end{equation}

\noindent where the dimensionless constants $g$ and $g_{N}$ are the electronic
and nuclear $g$ factors respectively; $\beta$ and $\beta_{N}$ are electronic
Bohr and nuclear magnetons.  The vectors {\bf I}, ${\bf s_{i}}$ and ${\bf
r_{i}}$ represent the nuclear spin, electronic spin for electron $i$, and
electronic position for electron $i$ with respect to the nucleus, operators.
When Eq.\ (\ref{micro1}) is expanded the microscopic Hamiltonian becomes,

\begin{equation}
\hat{\cal H}_{\rm micro} = -\frac{\mu_{0}}{4\pi}g\beta g_{N}\beta_{N}
  \sum_{i=1}^{n}
  \left( \frac{1}{{r_{i}}^{3}}\right)
  \sum_{\tau,\upsilon}I_{\tau}s_{i\upsilon}
  \left( \frac{{r_{i}}^{2}\delta_{\tau\upsilon} - 3\tau_{i}\upsilon_{i}}
  {{r_{i}}^{2}} \right) .
\label{micro2}
\end{equation}

If we consider here a single determinant electronic wavefunction, $\Psi_{q}$,
{\it i.e.}, one for a given electronic configuration and labeled here with a
$q$, then we can form a product basis with the nuclear spin eigenfunctions
which
corresponded to the eigenkets given in Eq.\ (\ref{macro2}).  The resulting
basis
vectors are given by,

\begin{equation}
|q, M_{S}; I, M_{I}\rangle = |q, M_{S}\rangle |I, M_{I}\rangle .
\label{micro3}
\end{equation}

\noindent The basis functions corresponding to these kets are operated upon by
the microscopic Hamiltonian, Eq.\ (\ref{micro2}).

\section{Equating the Expectation Values}

In order to determine the anisotropic hyperfine coupling constants,
$A_{\tau\upsilon}$, we compare identical matrix elements of the macroscopic and
microscopic Hamiltonian matrices and equate like terms.  Here we concern
ourselves with only those terms diagonal in $S$ and $I$.  Thus we have,

\begin{eqnarray}
 & & \sum_{\tau\upsilon}A_{\tau\upsilon}\langle S, {M_{S}}'; I, {M_{I}}'|
  I_{\tau}S_{\upsilon} |S, {M_{S}}; I, {M_{I}}\rangle
\nonumber  \\
 & = & -\frac{\mu_{0}}{4\pi}g\beta g_{N}\beta_{N}
  \sum_{i=1}^{n} \langle q, {M_{S}}'; I, {M_{I}}'|
  \left( \frac{1}{{r_{i}}^{3}}\right)
\nonumber \\
 &   &  \times \sum_{\tau,\upsilon}I_{\tau}s_{i\upsilon}
  \left( \frac{{r_{i}}^{2}\delta_{\tau\upsilon} - 3\tau_{i}\upsilon_{i}}
  {{r_{i}}^{2}} \right) |q, M_{S}; I, M_{I}\rangle
\label{equate1}
\end{eqnarray}

\noindent so equating terms we obtain,

\begin{equation}
\langle S, {M_{S}}'|S_{\upsilon}|S, M_{S}\rangle A_{\tau\upsilon} =
  k \langle q, {M_{S}}'| \sum_{i=1}^{n} \left( \frac{1}{{r_{i}}^{3}}\right)
  \sum_{\tau,\upsilon}s_{i\upsilon}
  \left( \frac{{r_{i}}^{2}\delta_{\tau\upsilon} - 3\tau_{i}\upsilon_{i}}
  {{r_{i}}^{2}} \right) |q, M_{S}\rangle
\label{equate2}
\end{equation}

\noindent where,

\begin{equation}
k = -\frac{\mu_{0}}{4\pi}g\beta g_{N}\beta_{N}
\label{equate3}
\end{equation}

\noindent and we have facilitated this comparison by choosing eigenkets for
nuclear spin states which have nonzero $M_{I}$, and, of course, a nonzero $I$.

The nuclear dipole-electronic dipole interaction is described by the one
electron operator given in Eq.\ (\ref{micro1}).  When evaluating the
expectation
value of such an operator when the determinantal wavefunctions representing
the bras and kets do not differ, {\it i.e.}, when $q$ and $M_{S}$ are the same
in each bra and ket, the following expression can be shown to hold
[\ref{bib:sza82}].

\begin{eqnarray}
        |q\rangle & = & |\cdots lm\cdots\rangle
        \label{oneelec1} \\
        \langle q|\hat{{\cal O}}_{1}|q\rangle & = & \sum_{l}^{n}\langle
l|h|l\rangle
        \label{oneelec2}
\end{eqnarray}

\noindent where $\hat{{\cal O}}_{1}$ is the one electron operator and is given
by,

\begin{equation}
\hat{{\cal O}}_{1} = \sum_{i=1}^{n}h(i) .
\label{oneelec3}
\end{equation}

\noindent and the $l$ and $m$ labels denote the spinorbitals $\chi_{l}$ and
$\chi_{m}$.  From Eq.\ (\ref{oneelec1}) - (\ref{oneelec3}), we obtain for
Eq.\ (\ref{equate2}) [\ref{bib:bev71}],

\begin{equation}
A_{\tau \upsilon} = \frac{k}{2\langle S_{Z}\rangle}
  \sum_{\mu,\nu}^{N}\rho_{\mu\nu}
    \langle \mu | {r_{i}}^{- 5}({r_{i}}^{2}\delta_{\tau \upsilon} - 3\tau_{i}
     \upsilon_{i}) |\nu\rangle
\label{equate4}
\end{equation}

\noindent where $\mu$ and $\nu$ are basis functions and the double sum is taken
over all of them.  The $\rho_{\mu\nu}$ is known as the spin density matrix and
is given by the difference in the alpha and beta density matrices.

\begin{eqnarray}
        \rho_{\mu\nu} & = & {P_{\mu\nu}}^{\alpha} - {P_{\mu\nu}}^{\beta}
        \\
        {P_{\mu\nu}}^{\alpha} & = & \sum_{i=1}^{p}{c_{\mu i}}^{\alpha}
          {c_{\nu i}}^{\alpha}
        \\
        {P_{\mu\nu}}^{\beta} & = & \sum_{i=1}^{q}{c_{\mu i}}^{\beta}
          {c_{\nu i}}^{\beta}
\end{eqnarray}

\noindent where the alpha and beta sums are over the $p$ occupied alpha
orbitals
and the $q$ occupied beta orbitals respectively.  The $c_{\mu i}$ are the
molecular orbital expansion coefficients for basis functions $\mu$ and MO $i$.

Eq.\ (\ref{equate4}) is known as averaging the given orbital operator over spin
density.  It should be noted that the spin density matrix is intrinsicly
different to the charge density matrix, which is the sum of the alpha and beta
density matrices.  It is of interest that elements of the electric field
gradient are calculated in the same way as above except the average is taken
over charge density.

\section*{Bibliography}
\begin{enumerate}
\item \label{bib:car67} A. Carrington and A. D. McLachlan, {\it Introduction to
      Magnetic Resonance} (Harper and Row, New York, 1967).
\item \label{bib:bev71} D. L. Beveridge and J. W. McIver, Jr., J. Chem. Phys.
      {\bf 54}, 4681, (1971).
\item \label{bib:sza82} A. Szabo and N. S. Ostlund, {\it Modern Quantum
      Chemistry:  Introduction to Advanced Electronic Structure Theory}
      (MacMillian, New York, 1983).
\end{enumerate}

\end{document}

Fiona Bettens                 The Australian National University,
Email: [email-masked]   Department of Chemistry,
Fax:  61 6 249 0760           Canberra,  ACT,  0200, AUSTRALIA