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# pimplem (c40b1)

Implementation of the Thermodynamic Simulation Method

* Description | How Chemical Perturbation works.

* File Formats | Output File Formats for Chemical Perturbation.

* IC Implementation | Implementation and File Formats for Internal

Coordinate Perturbation

Top

How the Chemical Perturbation Energy Calculation Works

For thermodynamic perturbation calculations the atoms making up

the system described by the hybrid Hamiltonian, H(lambda), can be divided

into four groups. 1) The environment part - all atoms that do not change

during the perturbation. E.g., for ethanol -> propane the solvent and

the terminal methyl group. 2) The reactant atoms - the atoms that are

present at lambda = 0 and absent at lambda = 1. 3) The product atoms -

the atoms that are absent at lambda = 0 and present at lambda = 1. 4)

The COLO atoms - atoms that are present in both the reactant and product

but change charge in going from one to the other.

Certain basic premises underly our approach. Energy values are

factored by lambda (or functions thereof), never the energy functions

themselves. The standard energy routines are called unchanged and can be

modified without requiring changes to the perturbation routines as long

as the calling sequence remains the same. Potential energy terms are

written to output during a trajectory and in the case of the window

method trajectories can be combined. Futhermore any lambda -> lambda'

can be calculated post priori and additional lambda points can be added

as desired. Most other implementations do not appear to allow this.

There is, however, a price entailed namely a certain amount of redundant

calculation. Furthermore , purely as a matter of conceptual preference,

the entire perturbation part of the Hamiltonian is facter by lambda in

the same way. There has been some advocacy of factoring the attractive

and repulsive part of the Lennard-Jones potential with different powers

of lambda (see Cross).

We want to calculate the potential energy U(lambda) = Uenv +

(1-lambda)**N Ureac + lambda**N Uprod, where N is positive integer

exponent and Uenv is the energy of the common environment part of the

system. The residue topology file for the system undergoing the

perturbation has all the internal coordinate terms for both the product

and reactant parts and the regular CHARMM energy routine calculates an

energy term that in it's sum contains part of Ureac and Uprod along with

Uenv and in certain cases, as will be discussed shortly, an additional

term that needs to be removed. The residue description must contain

non-bonded exclusions between the product and reactant atoms. Of course,

none of this is factored correctly, or at all, by lambda.

The approach to obtaining a the correct U(lambda) is an indirect

one. Instead of making it so that the normal energy routine calculates

Uenv only and having the perturbation energy routine calcuated determine

(1-lambda)**N Ureac + lambda**N Uprod, we have it instead calculate the

amount that must be subtracted from the normal energy routine value (here

after referred to as Unorm) to get U(lambda). The previous statement

must be amended for the case where there are COLO atoms. Then, Unorm

contains a term that must be totally removed and is missing some terms

completely, which must be added.

For the internal coordinate energy terms and the non-bonded van

der Waals interactions, the amount that must be subtracted from Unorm to

obtain U(lambda) is given by:

U(lambda) = Unorm + Ucorr

since,

U(lambda) = U(env) + (1 - lambda)**N Ureac + lambda**N Uprod

and

Unorm = U(env) + Ureac + Uprod

then

-Ucorr = [1-(1-lambda)**N]Ureac + [1-lambda**N]Uprod .

We have currently ignored the electrostatic terms. If there are no COLO

atoms the above expressions hold true for those terms as well.

If there are COLO atoms , the situation becomes a bit more

complicated. To discuss this the following nomenclature is introduced:

[reac| reac,colo-r,env]

The expression above indicates the calculation of the electrostatic

interaction between reactant atoms and 1) other reactant atoms 2) COLO

atoms with the reactant energy charges and 3) with environment atoms.

Unorm contains the following electrostatic terms:

[reac| reac, colo-r, env] + [color | prod, colo-r, env] +

[prod | prod, env]

The term [ colo-r | prod ] must be removed in it's entirety (product

atoms do not interact with reactant charges (colo-r). And the missing

interactions involving colo-p (product) charges must be added (suitably

factored by lambda). To do this Ucorr must contain:

(1 - (1 - lambda)**N) { [reac | reac, colo-r, env] +

[colo-r | colo-r, env] }

+ (1 - lambda**N)[prod | prod, env] + 1[color|prod]

- lambda**N [colo-p | colo-p, prod, env]

Note that -Ucorr is passed from the perturbation energy routine, thus the

negative term (last one) actually adds what is totally missing from

Unorm. The electrostatic contribution to Ucorr is actually calculated in

an even more round-about fashion than that which is given above.

First both the van der Waal's and electrostatic interactions

involving reactant and product atoms with everything (except interactions

between reactant and product atoms) are calculated. The reactant colo-r

charges are used for this. This provides the term:

(1 -(1-lambda)**N)[reac | env, colo-r, reac ]

and

(1 - lambda**N)[prod | env, colo-r, prod ].

If there are no COLO atoms, this is all we need (absent the colo-r term

in the expressions). Otherwise, three more calculations, involving only

the electrostatic energy, are required. The first involves interactions

between colo-r charges with environment and other colo-r charges:

(1-(1-lambda)**N)[colo-r | env, colo-r]

Next the colo-r product atom electrostatic interaction is calculated and

factored by a function of lambda that compensates for the amount in the

second ([prod | colo-r ...] ) calculation. In that term,

1-lambda**N[prod | colo-r] is included so we must determine,

(lambda**N)[colo-r| prod]

(Since the quantity (1-lambda**N) is calculated once we actually use,

(1 - (1 - lambda**N))[colo-r| prod]

Following this the colo-r charges are exchanged, temporarily, for colo-p

and the last calculations is done. The final expression is:

-lambda**N [colo-p | prod, env, colo-p]

Which actually adds (see above) the missing interaction into the total

potential energy.

The colo-r charges are restored after this. The same procedure

is done for the image atom calculation.

It is obvious that some optimization of this method is

achievable. One possibility is that by sorting the atom list so that

COLO, reactant and product atoms appear at the top of the list in that

order, most of the non-bonded list checking can be avoided and the

copying of data structures on the heap eliminated. A more radical change

would be to edit the non-bonded lists so that the normal energy routine

calculates only Uenv and the perturbation routines calculated Ureac and

Uprod directly. The presence of the COLO atoms makes both procedures

more complicated. However, there does not appear to be a viable

alternative to the COLO atoms that is consistant with our approach.

How the Chemical Perturbation Energy Calculation Works

For thermodynamic perturbation calculations the atoms making up

the system described by the hybrid Hamiltonian, H(lambda), can be divided

into four groups. 1) The environment part - all atoms that do not change

during the perturbation. E.g., for ethanol -> propane the solvent and

the terminal methyl group. 2) The reactant atoms - the atoms that are

present at lambda = 0 and absent at lambda = 1. 3) The product atoms -

the atoms that are absent at lambda = 0 and present at lambda = 1. 4)

The COLO atoms - atoms that are present in both the reactant and product

but change charge in going from one to the other.

Certain basic premises underly our approach. Energy values are

factored by lambda (or functions thereof), never the energy functions

themselves. The standard energy routines are called unchanged and can be

modified without requiring changes to the perturbation routines as long

as the calling sequence remains the same. Potential energy terms are

written to output during a trajectory and in the case of the window

method trajectories can be combined. Futhermore any lambda -> lambda'

can be calculated post priori and additional lambda points can be added

as desired. Most other implementations do not appear to allow this.

There is, however, a price entailed namely a certain amount of redundant

calculation. Furthermore , purely as a matter of conceptual preference,

the entire perturbation part of the Hamiltonian is facter by lambda in

the same way. There has been some advocacy of factoring the attractive

and repulsive part of the Lennard-Jones potential with different powers

of lambda (see Cross).

We want to calculate the potential energy U(lambda) = Uenv +

(1-lambda)**N Ureac + lambda**N Uprod, where N is positive integer

exponent and Uenv is the energy of the common environment part of the

system. The residue topology file for the system undergoing the

perturbation has all the internal coordinate terms for both the product

and reactant parts and the regular CHARMM energy routine calculates an

energy term that in it's sum contains part of Ureac and Uprod along with

Uenv and in certain cases, as will be discussed shortly, an additional

term that needs to be removed. The residue description must contain

non-bonded exclusions between the product and reactant atoms. Of course,

none of this is factored correctly, or at all, by lambda.

The approach to obtaining a the correct U(lambda) is an indirect

one. Instead of making it so that the normal energy routine calculates

Uenv only and having the perturbation energy routine calcuated determine

(1-lambda)**N Ureac + lambda**N Uprod, we have it instead calculate the

amount that must be subtracted from the normal energy routine value (here

after referred to as Unorm) to get U(lambda). The previous statement

must be amended for the case where there are COLO atoms. Then, Unorm

contains a term that must be totally removed and is missing some terms

completely, which must be added.

For the internal coordinate energy terms and the non-bonded van

der Waals interactions, the amount that must be subtracted from Unorm to

obtain U(lambda) is given by:

U(lambda) = Unorm + Ucorr

since,

U(lambda) = U(env) + (1 - lambda)**N Ureac + lambda**N Uprod

and

Unorm = U(env) + Ureac + Uprod

then

-Ucorr = [1-(1-lambda)**N]Ureac + [1-lambda**N]Uprod .

We have currently ignored the electrostatic terms. If there are no COLO

atoms the above expressions hold true for those terms as well.

If there are COLO atoms , the situation becomes a bit more

complicated. To discuss this the following nomenclature is introduced:

[reac| reac,colo-r,env]

The expression above indicates the calculation of the electrostatic

interaction between reactant atoms and 1) other reactant atoms 2) COLO

atoms with the reactant energy charges and 3) with environment atoms.

Unorm contains the following electrostatic terms:

[reac| reac, colo-r, env] + [color | prod, colo-r, env] +

[prod | prod, env]

The term [ colo-r | prod ] must be removed in it's entirety (product

atoms do not interact with reactant charges (colo-r). And the missing

interactions involving colo-p (product) charges must be added (suitably

factored by lambda). To do this Ucorr must contain:

(1 - (1 - lambda)**N) { [reac | reac, colo-r, env] +

[colo-r | colo-r, env] }

+ (1 - lambda**N)[prod | prod, env] + 1[color|prod]

- lambda**N [colo-p | colo-p, prod, env]

Note that -Ucorr is passed from the perturbation energy routine, thus the

negative term (last one) actually adds what is totally missing from

Unorm. The electrostatic contribution to Ucorr is actually calculated in

an even more round-about fashion than that which is given above.

First both the van der Waal's and electrostatic interactions

involving reactant and product atoms with everything (except interactions

between reactant and product atoms) are calculated. The reactant colo-r

charges are used for this. This provides the term:

(1 -(1-lambda)**N)[reac | env, colo-r, reac ]

and

(1 - lambda**N)[prod | env, colo-r, prod ].

If there are no COLO atoms, this is all we need (absent the colo-r term

in the expressions). Otherwise, three more calculations, involving only

the electrostatic energy, are required. The first involves interactions

between colo-r charges with environment and other colo-r charges:

(1-(1-lambda)**N)[colo-r | env, colo-r]

Next the colo-r product atom electrostatic interaction is calculated and

factored by a function of lambda that compensates for the amount in the

second ([prod | colo-r ...] ) calculation. In that term,

1-lambda**N[prod | colo-r] is included so we must determine,

(lambda**N)[colo-r| prod]

(Since the quantity (1-lambda**N) is calculated once we actually use,

(1 - (1 - lambda**N))[colo-r| prod]

Following this the colo-r charges are exchanged, temporarily, for colo-p

and the last calculations is done. The final expression is:

-lambda**N [colo-p | prod, env, colo-p]

Which actually adds (see above) the missing interaction into the total

potential energy.

The colo-r charges are restored after this. The same procedure

is done for the image atom calculation.

It is obvious that some optimization of this method is

achievable. One possibility is that by sorting the atom list so that

COLO, reactant and product atoms appear at the top of the list in that

order, most of the non-bonded list checking can be avoided and the

copying of data structures on the heap eliminated. A more radical change

would be to edit the non-bonded lists so that the normal energy routine

calculates only Uenv and the perturbation routines calculated Ureac and

Uprod directly. The presence of the COLO atoms makes both procedures

more complicated. However, there does not appear to be a viable

alternative to the COLO atoms that is consistant with our approach.

Top

File Formats

This node provides information on the FES output file format.

The data file created during dynamics can only be written as an ASCII

formatted file. It starts with a title that is written using the

subroutine WRITITL and thus has the standard CHARMM title format. After

terminating the title with a line containing an asterisk in the first

column and nothing else, an information line follows, containing:

NSTEP, PERFRQ, NDEGF, NPUMB, LPOWER - 5(I6,1X)

The first two numbers are not currently used by the post-processor.

NDEGF, the number of degrees of freedom is used if the CTEMp flag is set.

Npumb is the number of umbrella dihedral angles. If the UMBR flag is set

in the PROCess command and NPUMB is non-zero, the umbrella sampling

correction will be effected. LPOWER is the exponent for lambda scaling.

Every PFREQ steps the FES information is written out the unit

specified in the SAVE statement. If umbrella sampling is invoked the

format is as follows:

NPRIV,AKMATI,LAMBDA,E,VPRTTR,VPRTTP,VPRTNR,VPRTNP,

VPRTVR,VPRTVP,VPRTER,VPRTEP,TOTE,TOTKE,PUMEP

FORMAT(I12,2(1X,1PG24.16E2),/,3(2(1PG24.16E2,1X),

1 1PG24.16E2,/),2(1PG24.16E2,1X),1PG24.16E2)

If umbrella sampling is not invoked it is as follows:

NPRIV,AKMATI,LAMBDA,E,VPRTTR,VPRTTP,VPRTNR,VPRTNP,

VPRTVR,VPRTVP,VPRTER,VPRTEP,TOTE,TOTKE

FORMAT(I12,2(1X,1PG24.16E2),/,3(2(1PG24.16E2,1X),

1 1PG24.16E2,/),1PG24.16E2,1X,1PG24.16E2)

Where:

NPRIV step number

AKMATI timestep in wierd CHARMM units

LAMBDA value of lambda at timestep

E total potential energy

VPRTTR V(reactant) potential energy

VPRTTP V(product) ""

VPRTNR V(reactant) potential energy vdw + electrostatic

VPRTNP V(product) ""

VPRTVR V(reactant) potential energy vdw

VPRTVP V(product) ""

VPRTER V(reactant) potential energy electrostatic

VPRTEP V(product) ""

TOTE Total energy (potential + kinetic)

TOTKE Total kinetic energy.

and with umbrella sampling:

PUMEP The exp[-beta(Vsur - Vact)] term.

File Formats

This node provides information on the FES output file format.

The data file created during dynamics can only be written as an ASCII

formatted file. It starts with a title that is written using the

subroutine WRITITL and thus has the standard CHARMM title format. After

terminating the title with a line containing an asterisk in the first

column and nothing else, an information line follows, containing:

NSTEP, PERFRQ, NDEGF, NPUMB, LPOWER - 5(I6,1X)

The first two numbers are not currently used by the post-processor.

NDEGF, the number of degrees of freedom is used if the CTEMp flag is set.

Npumb is the number of umbrella dihedral angles. If the UMBR flag is set

in the PROCess command and NPUMB is non-zero, the umbrella sampling

correction will be effected. LPOWER is the exponent for lambda scaling.

Every PFREQ steps the FES information is written out the unit

specified in the SAVE statement. If umbrella sampling is invoked the

format is as follows:

NPRIV,AKMATI,LAMBDA,E,VPRTTR,VPRTTP,VPRTNR,VPRTNP,

VPRTVR,VPRTVP,VPRTER,VPRTEP,TOTE,TOTKE,PUMEP

FORMAT(I12,2(1X,1PG24.16E2),/,3(2(1PG24.16E2,1X),

1 1PG24.16E2,/),2(1PG24.16E2,1X),1PG24.16E2)

If umbrella sampling is not invoked it is as follows:

NPRIV,AKMATI,LAMBDA,E,VPRTTR,VPRTTP,VPRTNR,VPRTNP,

VPRTVR,VPRTVP,VPRTER,VPRTEP,TOTE,TOTKE

FORMAT(I12,2(1X,1PG24.16E2),/,3(2(1PG24.16E2,1X),

1 1PG24.16E2,/),1PG24.16E2,1X,1PG24.16E2)

Where:

NPRIV step number

AKMATI timestep in wierd CHARMM units

LAMBDA value of lambda at timestep

E total potential energy

VPRTTR V(reactant) potential energy

VPRTTP V(product) ""

VPRTNR V(reactant) potential energy vdw + electrostatic

VPRTNP V(product) ""

VPRTVR V(reactant) potential energy vdw

VPRTVP V(product) ""

VPRTER V(reactant) potential energy electrostatic

VPRTEP V(product) ""

TOTE Total energy (potential + kinetic)

TOTKE Total kinetic energy.

and with umbrella sampling:

PUMEP The exp[-beta(Vsur - Vact)] term.

Top

Internal Coordinate Implementation and File Formats

We describe how we have incorporated the double-wide, multiple-point,

window method for computing conformational free energy surfaces into

internal coordinate constraint and perturbation code with other CHARMM

routines, and it also shows the order in which the tasks are carried out,

as well as the format of the perturbation data file.

The primary internal coordinate (i.c.) constraint, perturbation, and

post-processing commands, as well as other TSM commands, are parsed in the

subroutine TSMS. When an i.c. constraint command is read, TSMS calls

ICFSET to parse the remainder of the command and to set-up the data needed

for the constraint resetting algorithm. When an i.c. perturbation command

is read, TSMS calls ICPSET to parse the remainder of the command and to

set-up the data needed to do the i.c. perturbations. Post-processing

command parsing and set-up is handled by the subroutine TSMP.

Some time after the constraints and perturbations are specified, a

dynamics command is issued and the dynamics is set up. During the dynamics

set-up, a "header" is written to the i.c. perturbation data file (opened on

unit iunicp) using the following fortran write statement in the subroutine

DCNTRL:

write(iunicp,100) nicp,icpinc,ndegf,delta

100 format(3i6,f12.6)

The variable nicp is the number of internal coordinates that will be

perturbed, icpinc is the number of subintervals, ndegf is the number of

degrees of freedom, and delta is the timestep in AKMA units. After the

dynamics is set-up, DCNTRL calls DYNAMC to integrate the equations of

motion. The main dynamics loop in DYNAMC is summarized in the following

pseudo-fortran code:

do istep = istart,istop * loop over number of steps

...

call ENERGY * get U(z) and forces

take unconstrained dynamics step

call SHAKEA * satisfy shake and i.c. cons.

...

call DYNICP * do pert. and get int. EÕs;

...

end do

The subroutine ENERGY calculates the total potential energy and the forces

needed to propagate the dynamics. After an unconstrained dynamics step is

taken, SHAKEA is called to satisfy the SHAKE and i.c. constraints in an

iterative fashion. We have to iterate the SHAKE and i.c. constraints

together, because the SHAKE constraint resetting may cause an ic constraint

to be violated, and vice versa. This constraint resetting procedure is

illustrated with the following pseudo-code:

do while (.not.done) * e.g. until shake has converged

perform iteration of shake cons. resetting

call icfcns * satisfy i.c. constraints

done = done.and..not.anyadj * if any i.c. constraints

* were adjusted, anyadj =

.true.

end do while

After the constraints are satisfied, DYNICP is called to do the double-

wide, multiple-point perturbations and calculate the interaction energies.

In DYNICP, the subroutine EIICP is called first to compute the interaction

energy of the unperturbed system (esbnp). Then the internal coordinate

values are obtained and some data is written to the perturbation data file:

call EIICP * get E for unperturbed system (esbnp)

do i = 1,nicp

get icval(1,i) * get unperturbed i.c. values

end do

c write data for unperturbed system:

write(iunicp,100) npriv,akmati,tote,totke,esbnp

100 format(i7,f10.4,3d16.8)

The two-dimensional array icval holds the internal coordinate values. The

unperturbed values are held in the first row, the values from the forward

perturbation in the second row, and the values from the reverse

perturbation in the third row. The data written to the data file includes

the number of the current dynamics step (npriv), the current simulation

time in AKMA units (akmati), the total energy (tote), total kinetic energy

(totke), and the interaction energy of the unperturbed system (esbnp).

Next, the unperturbed coordinates are copied into temporary arrays so

they can be restored after the perturbations have been carried out. Then

the double-wide, multiple point perturbations are carried out in a loop

over subintervals. The forward perturbation in each subinterval is done

first, followed by the reverse perturbation. The subroutine MVICP moves

the atoms involved in the perturbations using the algorithms described

above, and EIICP computed the interaction energies. The following pseudo-

code shows how these tasks are dispatched:

copy coords. into temp. arrays

scale = 0.0

dscale = 1.0/icpinc

do inc = 1,icpinc * loop over subintervals

scale = scale + dscale

call mvicp * move atoms by scale*dz

call eiicp * get int. E for forward pert. (esbfp)

do i = 1,nicp

get icval(2,i) * get perturbed i.c. values

end do

restore coords. from temp. arrays

call mvicp * move atoms by Ðscale*dz

call eiicp * get E for reverse pert. (esbrp)

do i = 1,nicp

get icval(3,i) * get perturbed i.c. values

end do

After all of the atoms have been moved and the interaction energies have

been computed for the forward and reverse perturbations in a subinterval,

the interaction energies and internal coordinate values are written to the

data file, and the unperturbed coordinates are restored in preparation for

the next subinterval (or the next dynamics step):

c write interaction energies of perturbed systems

write(iunicp,101) scale,esbfp,esbrp

101 format(7x,f10.4,2d16.8)

c write internal coordinate values

do i = 1,nicp

write(iunicp,102)

ic,icptyp(i),icval(1,i),icval(2,i),icval(3,i)

102 format(9x,2i4,3d16.8)

end do

restore coordinates from temp. arrays

end do

Internal Coordinate Implementation and File Formats

We describe how we have incorporated the double-wide, multiple-point,

window method for computing conformational free energy surfaces into

internal coordinate constraint and perturbation code with other CHARMM

routines, and it also shows the order in which the tasks are carried out,

as well as the format of the perturbation data file.

The primary internal coordinate (i.c.) constraint, perturbation, and

post-processing commands, as well as other TSM commands, are parsed in the

subroutine TSMS. When an i.c. constraint command is read, TSMS calls

ICFSET to parse the remainder of the command and to set-up the data needed

for the constraint resetting algorithm. When an i.c. perturbation command

is read, TSMS calls ICPSET to parse the remainder of the command and to

set-up the data needed to do the i.c. perturbations. Post-processing

command parsing and set-up is handled by the subroutine TSMP.

Some time after the constraints and perturbations are specified, a

dynamics command is issued and the dynamics is set up. During the dynamics

set-up, a "header" is written to the i.c. perturbation data file (opened on

unit iunicp) using the following fortran write statement in the subroutine

DCNTRL:

write(iunicp,100) nicp,icpinc,ndegf,delta

100 format(3i6,f12.6)

The variable nicp is the number of internal coordinates that will be

perturbed, icpinc is the number of subintervals, ndegf is the number of

degrees of freedom, and delta is the timestep in AKMA units. After the

dynamics is set-up, DCNTRL calls DYNAMC to integrate the equations of

motion. The main dynamics loop in DYNAMC is summarized in the following

pseudo-fortran code:

do istep = istart,istop * loop over number of steps

...

call ENERGY * get U(z) and forces

take unconstrained dynamics step

call SHAKEA * satisfy shake and i.c. cons.

...

call DYNICP * do pert. and get int. EÕs;

...

end do

The subroutine ENERGY calculates the total potential energy and the forces

needed to propagate the dynamics. After an unconstrained dynamics step is

taken, SHAKEA is called to satisfy the SHAKE and i.c. constraints in an

iterative fashion. We have to iterate the SHAKE and i.c. constraints

together, because the SHAKE constraint resetting may cause an ic constraint

to be violated, and vice versa. This constraint resetting procedure is

illustrated with the following pseudo-code:

do while (.not.done) * e.g. until shake has converged

perform iteration of shake cons. resetting

call icfcns * satisfy i.c. constraints

done = done.and..not.anyadj * if any i.c. constraints

* were adjusted, anyadj =

.true.

end do while

After the constraints are satisfied, DYNICP is called to do the double-

wide, multiple-point perturbations and calculate the interaction energies.

In DYNICP, the subroutine EIICP is called first to compute the interaction

energy of the unperturbed system (esbnp). Then the internal coordinate

values are obtained and some data is written to the perturbation data file:

call EIICP * get E for unperturbed system (esbnp)

do i = 1,nicp

get icval(1,i) * get unperturbed i.c. values

end do

c write data for unperturbed system:

write(iunicp,100) npriv,akmati,tote,totke,esbnp

100 format(i7,f10.4,3d16.8)

The two-dimensional array icval holds the internal coordinate values. The

unperturbed values are held in the first row, the values from the forward

perturbation in the second row, and the values from the reverse

perturbation in the third row. The data written to the data file includes

the number of the current dynamics step (npriv), the current simulation

time in AKMA units (akmati), the total energy (tote), total kinetic energy

(totke), and the interaction energy of the unperturbed system (esbnp).

Next, the unperturbed coordinates are copied into temporary arrays so

they can be restored after the perturbations have been carried out. Then

the double-wide, multiple point perturbations are carried out in a loop

over subintervals. The forward perturbation in each subinterval is done

first, followed by the reverse perturbation. The subroutine MVICP moves

the atoms involved in the perturbations using the algorithms described

above, and EIICP computed the interaction energies. The following pseudo-

code shows how these tasks are dispatched:

copy coords. into temp. arrays

scale = 0.0

dscale = 1.0/icpinc

do inc = 1,icpinc * loop over subintervals

scale = scale + dscale

call mvicp * move atoms by scale*dz

call eiicp * get int. E for forward pert. (esbfp)

do i = 1,nicp

get icval(2,i) * get perturbed i.c. values

end do

restore coords. from temp. arrays

call mvicp * move atoms by Ðscale*dz

call eiicp * get E for reverse pert. (esbrp)

do i = 1,nicp

get icval(3,i) * get perturbed i.c. values

end do

After all of the atoms have been moved and the interaction energies have

been computed for the forward and reverse perturbations in a subinterval,

the interaction energies and internal coordinate values are written to the

data file, and the unperturbed coordinates are restored in preparation for

the next subinterval (or the next dynamics step):

c write interaction energies of perturbed systems

write(iunicp,101) scale,esbfp,esbrp

101 format(7x,f10.4,2d16.8)

c write internal coordinate values

do i = 1,nicp

write(iunicp,102)

ic,icptyp(i),icval(1,i),icval(2,i),icval(3,i)

102 format(9x,2i4,3d16.8)

end do

restore coordinates from temp. arrays

end do