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# 1. Data Frame

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The data frame provides the user with a choice of predetermined potential energy functions or reads the potential from a data file or the windows clipboard. The choices are:

#### 1.a.  Morse Potential

V(r) = De (1-exp(-b (r-re)))2

De is the well depth, re is the location of the well minimum, and b controls the width. For a morse potential the energy levels E(v) are described by:

E(v) = w e (v+1/2) - w exe (v+1/2)2

v is the vibrational quantum number, w e is the harmonic vibrational frequency and is related to b and De by:

we = b ( De h /(2p2 cm ))1/2

h is Planck's constant, c is the speed of light and m is the mass of the particle in the well. For we and De in cm-1, b in Å-1, and m; in amu this can be written as:

we = b ( De 67.4305 / m))1/2

wexe is the anharmonic vibrational constant and is related to b and De by:

wexe = hb2/ (8p2cm)

For the same units as above this can be written as:

wexe = 16.8576 b2/p

See "Molecular Spectra and Molecular Structure 1. Spectra of Diatomic Molecules" G. Herzberg, 1950, D. Van Nostrand Co., New York, for further details.

The constants De, re, and b are set in the Parameters Frame at the lower left of the main window.

An approximation to the vibrational levels of HF can be described using Morse parameters:

re = 0.916808 Å
b = 2.259 Å-1
De = 47634.8 cm-1
mass = 0.957 amu.

A range of 0.5 to 2.5 Å and 60 points provides a good display. The first few energy levels should be:

v

E(v)

E(v)-E(0)

E9V)-E(v-1)

0

2046.8

1

6005.6

3958.7897

3958.7897

2

9784.6

7737.7972

3779.0075

3

13383.8

11337.0225

3599.2253

#### 1.b. Double Well Potential

This is actually used to describe any potential of the form:

V(r) = C2 r2 + C4 r4 + C6 r6 + C8 r8 + C10 r10 + C12 r12

The coefficients Cn are set in the Parameter frame.

This can be used to describe potentials with a double minimum such as the inversion motion in ammonia (NH3). An approximation of the ammonia inversion potential is given by:

C2 = -22000 cm-1 Å-2
C4 = 61000 cm-1 Å-4
mass = 2.47 amu.

A range of -1.0 to +1.0 Å and 60 points provides a good display.

The first few energy levels should be:

v

E(v)

E(v)-E(0)

E(v)-E(v-1)

0

527.0

1

527.5

0.5378

0.5378

2

1460.1

933.1356

932.5978

3

1493.8

966.8211

33.6854

4

2095.9

1568.9740

602.1530

5

2390.5

1863.5367

294.5627

6

2885.7

2358.7831

495.2464

#### 1.c. cos(nx) Potential

This can be used to describe potentials of the form:

V(x) = 1/2(V0 + V1 cos(x) + V2 cos(2x) + V3 cos(3x) + V4 cos(4x) + V6 cos(6x) + V12 cos(12x))

For the cos(nx) potential the range is set automatically to be 0 to 2p (p-1)/p where p is the number of points in the grid (set in the Points box in the Range frame). For this potential x is assumed to be an angle in radians. This requires a moment of inertia (in amu Å2) to be used instead of a mass.

The cos(nx) potential is useful for describing internal rotations. An approximation of the Ethane (CH3CH3) internal rotation potential is given by:

V3 = 1024 cm-1

mass (moment of inertia) = 1.577 amu Å2

The range is set automatically. 60 points provide a good display. The first few energy levels should be:

v

E(v)

E(v)-E(0)

E(v)-E(v-1)

0

150.7

1

150.7

0.0054

0.0054

2

150.7

0.0054

0.0000

3

438.0

287.3741

287.3686

4

438.0

287.3741

0.0000

5

438.3

287.5988

0.2247

6

692.1

541.4314

253.8326

One of the nice features of the FGH method is that it has periodic boundary conditions. This makes it ideal for internal rotations with finite barriers, as the wavefunctions naturally go over to free rotation wavefunctions above the barrier.

This allows the program to get data from an external text file. The x and y values can be delimited with spaces, commas or tabs. An example text file looks like:

 21 0.7 20886.8 0.8 4374.7 0.9 33.1 1 1747.8 1.1 6400.2 1.2 12354.6 1.3 18741.5

The first line which is the number of points. Subsequent lines are x y pairs in units of Å for x and cm-1 for y. For rotational potentials the units are radians for x and cm-1 for y (in which case the mass should be entered as a moment of inertia in units of amu Å2).

The program does not understand scientific notation. The program reports the range and the number of points in the file in the Parameters frame. The program performs a cubic spline through the points to generate the points used by the FGH algorithm. The number of points and range can be adjusted in the Range frame. Setting the range outside of the range of points in the data file can lead to random potentials as the spline fit may not extrapolate well.