# Hessians¶

## Overview¶

This section relates to the computation of analytic hessians which are available for open and closed shell SCF, except ROHF and for closed shell and unrestricted open shell DFT [1]. Analytic hessians are not currently available for SCF or DFT calculations relativistic all-electron methodologies or for charge fitting with DFT. The current algorithm is fully in-core and does not use symmetry.

There is no required input for the Hessian module. This module only impacts the hessian calculation. For options for calculating the frequencies, please see the Vibrational module.

## Hessian Module Input¶

All input for the Hessian Module is optional since the default definitions are usually correct for most purposes. The generic module input begins with hessian and has the form:

 hessian
thresh <real tol default 1d-6>
print ...
profile
end


### Defining the wavefunction threshold¶

You may modify the default threshold for the wavefunction. This keyword is identical to THRESH in the SCF, and the CONVERGENCE gradient in the DFT. The usual defaults for the convergence of the wavefunction for single point and gradient calculations is generally not tight enough for analytic hessians. Therefore, the hessian, by default, tightens these up to 1d-6 and runs an additional energy point if needed. If, during an analytic hessian calculation, you encounter an error:

 cphf_solve:the available MOs do not satisfy the SCF equations


the convergence criteria of the wavefunction generally needs to be tightened.

### Profile¶

The PROFILE keyword provides additional information concerning the computation times of different sections of the hessian code. Summary information is given about the maximum, minimum and average times that a particular section of the code took to complete. This is normally only useful for developers.

Known controllable print options are shown in the table below:

Name Print Level Description
“cphf_cont” debug detailed CPHF information
“nucdd_cont” debug detailed nuclear contribution information
“onedd_cont” debug detailed one electron contribution information
“twodd_cont” debug detailed two electron contribution information
“fock_xc” debug detailed XC information during the fock builds

Hessian Print Control Specifications

# Vibrational frequencies¶

The nuclear hessian which is used to compute the vibrational frequencies can be computed by finite difference for any ab initio wave-function that has analytic gradients or by analytic methods for SCF and DFT (see Hessians for details). The appropriate nuclear hessian generation algorithm is chosen based on the user input when TASK frequencies is the task directive.

The vibrational package was integrated from the Utah Messkit and can use any nuclear hessian generated from the driver routines, finite difference routines or any analytic hessian modules. There is no required input for the “VIB” package. VIB computes the Infra Red frequencies and intensities for the computed nuclear hessian and the “projected” nuclear hessian. The VIB module projects out the translations and rotations of the nuclear hessian using the standard Eckart projection algorithm. It also computes the zero point energy for the molecular system based on the frequencies obtained from the projected hessian.

The default mass of each atom is used unless an alternative mass is provided via the geometry input or redefined using the vibrational module input. The default mass is the mass of the most abundant isotope of each element. If the abundance was roughly equal, the mass of the isotope with the longest half life was used.

In addition, the vibrational analysis is given at the default standard temperature of 298.15 degrees.

## Vibrational Module Input¶

All input for the Vibrational Module is optional since the default definitions will compute the frequencies and IR intensities. The generic module input can begin with vib, freq, frequency and has the form:

 {freq || vib || frequency}
[reuse [<string hessian_filename>]]
[mass <integer lexical_index> <real new_mass>]
[mass <string tag_identifier> <real new_mass>]
[{temp || temperature} <integer number_of_temperatures>\
<real temperature1 temperature2 ...>]
[animate [<real step_size_for_animation>]]
[fd_delta [<real step_size_for_fd_hessian>]]
[filename <string file_set_name> [overwrite]]
end


### Hessian File Reuse¶

By default the task frequencies directive will recompute the hessian. To reuse the previously computed hessian you need only specify reuse in the module input block. If you have stored the hessian in an alternate place you may redirect the reuse directive to that file by specifying the path to that file.

 reuse /path_to_hessian_file


This will reuse your saved Hessian data but one caveat is that the geometry specification at the point where the hessian is computed must be the default “geometry” on the current run-time-data-base for the projection to work properly.

### Redefining Masses of Elements¶

You may also modify the mass of a specific center or a group of centers via the input.

To modify the mass of a specific center you can simply use:

 mass 3 4.00260324


which will set the mass of center 3 to 4.00260324 AMUs. The lexical index of centers is determined by the geometry object.

To modify all Hydrogen atoms in a molecule you may use the tag based mechanism:

 mass hydrogen 2.014101779


The mass redefinitions always start with the default masses and change the masses in the order given in the input. Care must be taken to change the masses properly. For example, if you want all hydrogens to have the mass of Deuterium and the third hydrogen (which is the 6th atomic center) to have the mass of Tritium you must set the Deuterium masses first with the tag based mechanism and then set the 6th center’s mass to that of Tritium using the lexical center index mechanism.

The mass redefinitions are not fully persistent on the run-time-data-base. Each input block that redefines masses will invalidate the mass definitions of the previous input block. For example,

freq
reuse
mass hydrogen 2.014101779
end
freq
reuse
mass oxygen 17.9991603
end


will use the new mass for all hydrogens in the first frequency analysis. The mass of the oxygen atoms will be redefined in the second frequency analysis but the hydrogen atoms will use the default mass. To get a modified oxygen and hydrogen analysis you would have to use:

freq
reuse
mass hydrogen 2.014101779
end
freq
reuse
mass hydrogen 2.014101779
mass oxygen 17.9991603
end


### Temp or Temperature¶

The “VIB” module can generate the vibrational analysis at various temperatures other than at standard room temperature. Either temp or temperature can be used to initiate this command.

To modify the temperature of the computation you can simply use:

 temp 4 298.15 300.0 350.0 400.0


At this point, the temperatures are persistant and so the user must “reset” the temperature if the standard behavior is required after setting the temperatures in a previous “VIB” command, i.e.

 temp 1 298.15


### Animation¶

The “VIB” module also can generate mode animation input files in the standard xyz file format for graphics packages like RasMol or XMol There are scripts to automate this for RasMol in \$NWCHEM_TOP/contrib/rasmolmovie. Each mode will have 20 xyz files generated that cycle from the equilibrium geometry to 5 steps in the positive direction of the mode vector, back to 5 steps in the negative direction of the mode vector, and finally back to the equilibrium geometry. By default these files are not generated. To activate this mechanism simply use the following input directive

 animate


anywhere in the frequency/vib input block.

Given an ordered list of files containing molecular coordinates in XYZ format, the rasmolmovie shell script generates an animated gif for each of the six possible views down a Cartesian axis.

It uses the free utilities

It should be easy to modify the script to other file formats or animation tools.

#### Controlling the Step Size Along the Mode Vector¶

By default, the step size used is 0.15 a.u. which will give reliable animations for most systems. This can be changed via the animate input directive, e.g.

vib
animate 0.20
end


where is the real number that is the magnitude of each step along the eigenvector of each nuclear hessian mode in atomic units.

### Specifying filenames for animated normal modes¶

By default, normal modes will be stored in files that start with “freq.m-“. This is inconvenient if more than vibrational analysis is run in a single input file. To specify different filename for a particular vibrational analysis use the directive

filename <file_set_name> [overwrite]


where is the name that will be prepended to the usual filenames. In addition the code by default requires all files to be new files. When the option “overwrite” is provided any pre-existing files will simply be overwritten.

### Controlling the Step Size of the Finite difference Hessian¶

By default, the step size used for calculating the finite difference Hessian is 0.010 a.u. for DFT and NWPW modules, and 0.001 a.u. otherwise This can be changed via the fd_delta input directive, e.g.

vib
fd_delta 0.005
end


where is the real number that is the magnitude of each displacement in atomic units for the calculation of the finite difference Hessian. For older versions of NWChem without the fd_delta option just set the “stpr_gen:delta” value on the runtime database, e.g.

set stpr_gen:delta 0.005


### An Example Input Deck¶

This example input deck will optimize the geometry for the given basis set, compute the frequencies for H2O, H2O at different temperatures, D2O, HDO, and TDO.

start  h2o
title Water
geometry units au autosym
O      0.00000000    0.00000000    0.00000000
H      0.00000000    1.93042809   -1.10715266
H      0.00000000   -1.93042809   -1.10715266
end
basis noprint
H library sto-3g
O library sto-3g
end
scf; thresh 1e-6; end
driver; tight; end

scf; thresh 1e-8; print none; end

freq
reuse; temp 4 298.15 300.0 350.0 400.0
end

freq
reuse; mass H 2.014101779
temp 1 298.15
end

freq
reuse; mass 2 2.014101779
end

freq
reuse; mass 2 2.014101779 ; mass 3 3.01604927
end
`

## References¶

1. Johnson, B.G. and Frisch, M.J. (1994) “An implementation of analytic second derivatives of the gradient-corrected density functional energy”, Journal of Chemical Physics 100 7429-7442, DOI:10.1063/1.466887