Molecule Types Vibrational Structure | <Prev Next> |

The force field can be expressed in terms of valence or
symmetry coordinates. A typical analysis will include several
isotopologues and might have a data file structured something
like this:

Note the various single and multiple objects:

- A single Global Variables object
which contains geometric variables and force constants common
to all isotopologues.

- A z Matrix for each isotopologue, which specifies the geometry. The use of variables means this will be identical for all isotopologues, except for the masses.
- An (optional) Internal Coordinates object for each isotopologue, which specifies the valence coordinates and force constants. The use of variables means this will be identical for all isotopologues.
- An (optional) Symmetry Coordinates object for each isotopologue, which specifies the combinations of valence coordinates to use. The use of variables means this will only differ for isotopologues of different symmetry.

A Cartesian Coordinates
object may also be present; the exact combination of objects
used depends on the required calculation. Nuclear Co-ordinates and Vibrational Modes objects will
also be present, automatically created and updated from the
force field analysis objects listed above.

The current version of the program does not explicitly make
use of symmetry for the force field analysis, but it is implicit
in the way the force constants and geometry are specified. The
current version of PGOPHER is restricted to harmonic terms, but
the program is structured so that anhamonic terms can be added
in future. The units are currently less flexible than for
rotational calculations; the overall mixture units must be set
to `cm``-1`, the geometry specified in Angstroms
and degrees and force constants (for stretches) in
millidyne/Å = aJ/Å^{2} (1 millidyne/Å
= 1 aJ/Å^{2} = 100 J/m^{2} = 100 N/m). For
a full discussion of the units see Hedberg and Mills (1993).

The 3*N*-6 (or more) internal coordinates, **R**,
are defined in terms of a matrix, **B**, that transforms
between the 3*N* Cartesian displacements, **Δx**
and **R**:

R=BΔx

**B** can be specified directly with a Cartesian Coordinates object, or
in terms of Valence Coordinates
with an Internal Coordinates
object. The potential energy can be specified in terms of internal
co-ordinates:

The implementation permits redundant internal coordinates (i.e. more than 3V= ½R^{T}fR

Given

whereρ=M^{½}Δx=lQ

With this definition ofΔx=M^{-½}lQ=adm Q

2where ' is used to indicate d/dT=ρ'ρ' =Q'^{T}l^{T}' =lQQ'^{T}Q' = Σ_{i}Q_{i}'^{2}

2This can be rewritten in terms of normal coordinates usingV=ρ^{T}M^{-½ }B^{T}fBM^{-½}ρ

2where theV=Q^{T}l^{T }M^{-½ }B^{T}fBM^{-½}lQ=Q^{T}l^{T }GFlQ

TheGF=M^{-½ }B^{T}fBM^{-½}

l^{T }GF=lλ

where **λ **is a diagonal matrix,
and the diagonal elements are the force constants in the normal
coordinates, **Q**. Note that the 5 or 6 lowest frequency
modes will correspond to translations and rotations, and will be
excluded from most subsequent calculations. The test for this is
on λ; values smaller than `SmallCoefficient`
of the largest are discarded. The potential energy thus has the
required diagonal form:

2V=Q^{T}λQ= Σ_{i}λ_{i}Q_{i}^{2}

The dimensionless normal coordinates are
related to the normal coordinates by a scaling factor**, **γ_{i}^{1/2}:

_{i} =
2πλ_{i}^{1/2}/*h* the total
energy has a particularly simple form:

*q*_{i} dimensionless as required
and identifies λ_{i}^{½} as
2πν_{i}. The transformation matrix between
these dimensionless normal coordinates and Cartesian
displacements, which we define as **d**, is then given by:

**d**^{int},
follows from this:

For more control, over the mode numbering for example, the internal
coordinates can be transformed to symmetry coordinates, If the scaling factor is chosen to be γq= γ_{i}_{i}^{1/2}Q_{i }

T+V= ½ Σ(_{i}P_{i}^{2}+ λ_{i}Q_{i}^{2}) = ½ Σ(2π)_{i}^{-1}hλ_{i}^{½}(p_{i}^{2}+q_{i}^{2}) = ½ Σ_{i}hν_{i}(p_{i}^{2}+q_{i}^{2})

which also makes

For a harmonic oscillator this the range of the zero point motion is ±1, so the elements of this matrix give the range of the zero point motion. The transformation matrix between dimensionless normal coordinates and internal coordinates,Δx=M^{-½}lQ=M^{-½}lγ^{-}^{½ }q =d q

R=BΔx =BM^{-½}lγ^{-}^{½}q=Bd q=d^{int}q

S=U R=U BΔx

The matrix **U **is specified in the Symmetry Coordinates object.
The potential energy can also be expressed in terms of symmetry
coordinates:

V= ½S^{T}FS

Note the use of **F** for the matrix of
force constants expressed in symmetry coordinates; it is related
to the force constant matrix, **f**, used above by:

f=U^{T}FU

Either** f** or **F** can be given but
the calculation of normal modes from symmetry coordinates requires
**F**. If non redundant coordinates are used, **U** will be
square so the above equation can be inverted by working out **U**^{-1}.
If redundant coordinates are used, they must be chosen such that **UU**^{T}
= **1**, in which case **F** = **U** **f**** ****U**^{T}.

When using symmetry coordinates a slightly different approach is used to find the normal modes. The (standard)**G** matrix is
calculated from:

**G**
matrix such that:

**C** is a lower triangular matrix. The matrix **C**^{T}**
F ****C** has the same eigenvalues, **λ**,
as the **GF** matrix, and the eigenvectors, **V**, can be
used to calculate the transformation matrix, **L**, between
the symmetry and normal coordinates:

**d**^{sym}, between
the symmetry and dimensionless normal coordinates:

**L** the following can be calculated:

**A** is the matrix that transforms symmetry coordinates
to Cartesian coordinates:

**A L** = **M**^{-1}**B**^{T}(**L**^{-1})^{T}
is calculated as the **l** matrix can be calculated
from:

When using symmetry coordinates a slightly different approach is used to find the normal modes. The (standard)

and Choleski decomposition is used to factorise theG=U BM^{-1}B^{T}U^{T }

whereG=C C^{T}

and the transformation matrix,S=CVQ = L Q

GivenS=L Q = Lγ^{-}^{½}q=d^{sym}q

where(L^{-1})^{T}=FLλ^{-1}

G^{-1}= (L^{-1})^{T}L^{-1 }A=M^{-1}B^{T}G^{-1}=M^{-1}B^{T}(L^{-1})^{T}L^{-1 }

In practice the matrixΔx=A S

=lM^{½}AL

- Worked Example - Setting up a
simple force field analysis for H
_{2}O - Worked Example - Setting up a
combined force field analysis for H
_{2}O and D_{2}O - Worked Example - Refining an
ab initio force field for H
_{2}O - z Matrix
- Internal Coordinates
- Valence Coordinates
- Cartesian Coordinates
- Symmetry Coordinates
- Sample Data Files

- L. Hedberg and I. M. Mills, J. Molec. Spectrosc.
**160**, 117 (1993). - L. Hedberg and I. M. Mills, J. Molec. Spectrosc.
**203**, 82 (2000).