Nonbonded interactions (frequently abreviated “nbond”) refer to
van der Waals terms and the electrostatic terms between all atom pairs
that are not specifically excluded from nonbond calculations
as for example are directly bonded atoms *nbx*.
These terms are defined on atom pairs and to a first aproximation would
require the number of atoms squared calculations. To avoid this burden
various truncation and approximation schemes can be employed in the
program, breaking the nonbonded calculation into two parts,
initialization and actual energy calculation.

The method of approximation, cutoffs, and other relevant parameters can be entered any time the nbond specification parser is invoked. See the syntax section for a list of all commands that invoke this parser.

Simple Ewald function is modified so it works with any shape of the simulation box. Comments/suggestions at hkamberaj@asu.edu

```
{ NBONds } { [INBFrq integer] nonbond-spec }
{ UPDAte ... } { }
{ ENERgy ... } { }
{ MINImize ... } { }
{ DYNAmics ... } { }
{ VIBRAN ... } { }
{ CORRel ... } { }
```

Note

The INBFrq value is remembered. If its value is zero, no interpretation of [nonbond-spec] will be made, as well as no modifications of the nonbond lists. It’s default value is -1 .

In all cases as many keywords and values as desired may be specified. The keywords are:

```
nonbond-spec::= [method-spec] [distances-spec] [misc-specs] [INIT] [RESET]
method-spec::= [ ELEC electrostatics-spec ] [ VDW vdw-spec ] [ BYCUbes ]
[ NOELectrostatics ] [ NOVDwaals ] [ BYGRoup ]
[ GRAPe grape-spec ] [ LIST ] [ BYCBim ]
[ NOGRape ] [ NOLIst ] [ BYCC ]
[ LRC ]
[ IPS ips-spec ]
electrostatics-spec::= [ EWALD ewald-spec ] [ FMA fma-spec ] elec-opt-spec
[ NOEWald ] [ NOFMA ] [ ACE ace-spec ]
[ EIPS ]
elec-opt-spec::=
[ ATOM ] [ CDIElec ] [ SHIFted ]
[ GROUp [ EXTEnded [GRADients] [QUADrip] ] ] [ RDIElec ] [ SWITched ]
[ [ [NOGRad ] [NOQUads] ] ] [ FSWItch ]
[ [ NOEXtended ] ] [ FSHIft ]
[ MSHIft ] ! MMFF
[ TRUNcate ] ! MMFF
vdw-spec::= [ VGROUP [ VSWITched ] ]
[ VATOM [ VSHIfted ] ]
[ [ VSWItched ] ]
[ [ VFSWitch ] ]
[ [ VTRUnc [ CTVT real ] ] ! MMFF only
[ [ VIPS ] ] ! NBIPS
ips-spec::= [RAIPS real] [RIPS real] [NIPSFRQ int] [DVBIPS real] -
[ [MIPSX int] [MIPSY int] [MIPSZ int] [MIPSO int]] -
[ [PXYZ] ]
[ [PXY|PYZ|PZX|PYX|PZY|PXZ] ]
[ [PX|PY|PZ] ]
distances-spec::= [general-dist] [warning-dist]
general-dist::=
[ CUTNB real ] [ CTONNB real ] [ CTOFNB real ] [ CTEXNB real ]
SOFT [EMAX real ] [MINE real] [MAXE real] VDWE ELEE ! SOFTVDW
warning-dist::=
[ WMIN real ] [ WRNMXD real ]
misc-specs::= [ EPS real ] [ E14Factor real ] [ NBXM integer] -
[ NBSCale real] [IMSCale real] [EXOForce] -
[ NORXN ]
[ RXNFLD rxnfld-spec ]
[ RXNNB rxnfld-spec ]
rxnfld-spec::= [ EPSEXT real ] [ ORDER integer ] [ SHELL real ]
fma-spec::= [LEVEL int] [TERMS int]
ewald-spec::= (see ewald::(chmdoc/ewald.doc)syntax )
ace-spec::= (see ace::(chmdoc/ace.doc)syntax )
```

The defaults for the nonbond specification reside with the parameter file. The defaults are specified at the begining of the van der Waal section. These defaults are the recommended options.

The following command contains all defaults for one of the
older protein parameter files, and is equvalent to the command `NBONds INIT`
in it usage when this parameter file is present.

```
NBONDS ELEC ATOM NOEX NOGR NOQU SWIT RDIE VATOM VDW VSWI -
CUTNB 8.0 CTEXNB 999.0 CTOFNB 7.5 CTONNB 6.5 WMIN 1.5 WRNMXD 0.5 -
EPS 1.0 NORXN EPSEXT 80.0 ORDER 10 SHELL 2.0 CTVTRN 10.0 -
E14FAC 1.0 NBSCAL 1.0 IMSCAL 1.0 NBXMOD 5 NOFMA -
NOEWALD NOPME KAPPA 1.0 KMAX 5 KMAXSQ 27 ERFMOD -1
```

MMFF specific defaults: `VTRUnc MSHFt E14Fac 0.75 CTVTrn 8.0`

SOFTVDW specific defaults:

```
SOFT EMAX 30.0/EPS MINE -300.0/EPS MAXE -2.*MINE ! CDIE
SOFT EMAX 15.0/EPS MINE -120.0/EPS MAXE -2.*MINE ! RDIE
```

Values do not change unless explicitly specified, except for the ON/OFF values which cascade when the cutoff values are changed as;

```
CTOFNB=CUTNB-0.5
CTONNB=CTOFNB-1.0
```

Warning

These old defaults have been shown to be detrimental to protein behavior. It is generally better to use the defaults in the parameter sets.

The presence of soft core nonbonded terms is recommended for calculations in a dense system (docking, loop refinement, NMR refinement). The initial value of RMIN (switching distance for the soft core potential) is recommended to be 0.885

RECOMMENDED:

Presented here is a suggested list of options. Where specifications are missing, substitute the defaults:

Use Isotropic Periodic Sum (IPS) calculation for either finite (in vacuum) or periodic systems:

`NBONDS IPS CUTNB 12.0 CTOFNB 10.0 EPS 1.0 CDIE`

or use IPS with fully homogenous assumption for either finite (in vacuume) or periodic systems:

`NBONDS IPS PXYZ CUTNB 12.0 CTOFNB 10.0 EPS 1.0`

or use IPS with 2D homogenous assumption for interfacial membrane systems:

`NBONDS IPS PXY CUTNB 12.0 CTOFNB 10.0 EPS 1.0`

or use VIPS for vdw and Ewald for charge interaction for periodic systems:

```
NBONDS VIPS -
ATOM EWALD PMEWALD KAPPA 0.32 FFTX 32 FFTY 32 FFTZ 32 ORDER 6 -
CUTNB 12.0 CTOFNB 10.0 EPS 1.0 CDIE
```

For no-cutoff periodic systems:

```
** ** **
NBONDS ATOM EWALD PMEWALD KAPPA 0.32 FFTX 32 FFTY 32 FFTZ 32 ORDER 6 -
CUTNB 12.0 CTOFNB 10.0 VDW VSHIFT
(** system size dependent - use about 0.8-1.0 grids per angstrom)
```

For atom based cutoffs:

```
NBONDS ATOM FSHIFT CDIE VDW VSHIFT -
CUTNB 13.0 CTOFNB 12.0 CTONNB 8.0 WMIN 1.5 EPS 1.0
```

or (perhaps better for longer cutoff distances, but more expensive)

```
NBONDS ATOM FSWITCH CDIE VDW VSHIFT -
CUTNB 13.0 CTOFNB 12.0 CTONNB 8.0 WMIN 1.5 EPS 1.0
```

For group based cutoffs (doesn’t vectorize well):

```
NBONDS GROUP FSWITCH CDIE VDW VSWITCH -
CUTNB 13.0 CTOFNB 12.0 CTONNB 8.0 WMIN 1.5 EPS 1.0
```

For extended electrostatics :

```
NBONDS GROUP SWITCH CDIE VDW VSWI EXTEND GRAD QUAD -
CUTNB 13.0 CTOFNB 12.0 CTONNB 8.0 WMIN 1.5 EPS 1.0
```

For a better description of these methods and how they perform, see: P.J. Steinbach, B.R. Brooks: “New Spherical-Cutoff Methods for Long-Range Forces in Macromolecular Simulation,” J. Comp. Chem. 15, 667-683 (1994).

OPTIONS THAT ARE NOT RECOMMENDED (OR REALLY BAD):

```
[ ATOM ] [ CDIElec ] [ SHIFted ] no (obsolete, but used in the past)
[ ATOM ] [ CDIElec ] [ SWITched ] NO! Very bad - do not use!
[ GROUp ] [ CDIElec ] [ SHIFted ] no (obsolete)
[ GROUp ] [ CDIElec ] [ SWITched ] NO! Very bad with non-neutral groups!
[ ATOM ] [ RDIElec ] [ SHIFted ] yes, but do you really want RDIE??
[ ATOM ] [ RDIElec ] [ SWITched ] no. switch is bad here.
```

The first time that the primary or image non-bond list is generated, an estimate is made, based on the number of atoms, of how much memory will be needed to store the pair list. If too large an estimate is made, memory will be wasted. If too small an estimate is made, a second (and larger) estimate will be made and the memory allocated on the first attempt is wasted. NBSCale is a correction factor to the initial estimate allowing better control of memory allocation. For example NBSCale 1.5 allocated 50% more memory than CHARMM would usually guess and NBSCale 0.8 allocated 20% less. IMSCale does the same thing when the image pair list is generated. The values of NBSCale and IMSCale must be determined empirically, but they can generate huge memory savings on large systems.

These keywords are valid wherever nonbond options may appear, e.g. ENERgy, DYNAmics, MINImiz, and UPDAte. Note that NBSCale must be used in the first statement which generates a nonbond list; an UPDAte without NBSCale followed by DYNAmics with NBSCale is ineffective.

For a system of about 17,000 atoms, a value of NBSCALE 1.5 was effective in providing about 25 MB of memory reduction compared to the the default NBSCale value (1.0); while for a system of about 10,000 atoms, the optimum of 1.4 gave a reduction of about 12 MB. For example:

```
mini abnr nstep 500 nprint 10 tolenr 1.e-6 cutnb 14. ctofnb 12. ctonnb 10. -
fshift cdie eps 1.0 vshift inbfrq 20 imgfrq 20 cutim 14. nbscale 1.5
```

Likewise for DNYAmics, ENERgy, or the UPDAte commands; the latter is useful in doing some trial and error probes to determine the optimum NBSCale value, with fixed sizes for CUTNB and CUTIM:

```
update cutnb 14. ctofnb 12. ctonnb 10. fshift cdie eps 1.0 vshift -
inbfrq 20 imgfrq 20 cutim 14. nbscale @1 -
```

where the csh or tcsh command line might be something like:

`% charmm medium 1:1.5 < tst_nbscal.inp >& nbscal_1.5 &`

The above is based on single processor calculations; the same general idea applies to parallel calculations, but the optimum value for NBSCale will be less than 1.0, perhaps 0.7 to 0.8 for systems in the 10K to 17K atom range. Memory usage can be further reduced using the IMSCale keyword; some experimentation will be required depending on the number of atoms and the cutoffs being used.

Update frequency for the non-bonded list. Used in the subroutine ENERGY() to decide whether to update the non-bond list. When set to :

0 no updates of the list will be done. +n an update is done every time MOD(ECALLS,n).EQ.0 . This is the old frequency-scheme, where an update is done every n steps of dynamics or minimization. -1 heuristic testing is performed every time ENERGY() is called and a list update is done if necessary. This is the default, because it is both safer and more economical than frequency-updating.

Every time the energy is called, the distance is computed each atom moved since the last list-update.

If any atom moved by more than (CUTNB - CTOFNB)/2 since the last list-update was done, then it is possible that some atom pairs in which the two atoms are now separated by less than CTOFNB are not in the pairs-list. So a list update is done.

If all atoms moved by less than (CUTNB - CTOFNB)/2 , then all atom pairs within the CTOFNB distance are already accounted for in the non-bond list and no update is necessary.

This section describes how programmers can control the list-updating behavior when their routines call the ENERGY() subroutine.

All list-updating decisions, whether they are frequency based or heuristic based, are made in the subroutine UPDECI(ECALLS) , which is called from only one place : at the very beginning of ENERGY().

UPDECI(ECALLS) can be controled through INBFRQ (via the CONTRL.FCM common block) and ECALLS (via the ENERGY.FCM common block) as follows :

If INBFRQ = +n non-bond list is performed when MOD(ECALLS,n).EQ.0 . Image and H-bond lists are updated according to IMGFRQ and IHBFRQ. If INBFRQ = 0 non-bond list update is not performed. Image and H-bond lists are updated according to IMGFRQ and IHBFRQ. If INBFRQ = -1 all lists are updated when necessary (heuristic test). (Note that ECALLS is incremented by ICALL every time ENERGY(,,,ICAL) is called. In most cases, ICALL=1)

The current implementation of UPDECI() will work (without modifications) to decide whether the image/crystal non-bond lists need updating, provided the periodicity parameters don’t change (i.e. constant Volume).

UPDECI() is easily adapted to variable Volume dynamics/minimizations. This is described in comments of the routine itself.

A programmer can sometimes skip the heuristic test itself, making the decision whether to do list-updating even more economical.

This option is only available if the size of the step taken since the last call to ENERGY() is known. For an example of usage, see the subroutine ENERG() in TRAVEL.

The electrostatic options are separate from the van der Waal options, though some keywords are shared between them. The following is a description of all options and keywords.

Electrostatics

The ELEC keyword (default) invokes electrostatics. The NOELec keyword suppresses all electrostatic energy terms and options. There are two basic methods for electrostatics, GROUp and ATOM. A model based on the GROUp method is the extended electrostatics model which approximates the full electrostatic interaction and eliminates the need for a cutoff function. This model is based on the partitioning of the electrostatic term into two contributions. One comes from the interaction between particles which are spatially close and is treated by conventional pairwise summation. The second contribution comes from interactions between particles which are spatially distant from one another and is treated by a multipole moment approximation.

[The original model was described in B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, M. Karplus. J. Comp. Chem., 4, 187, (1983) and more recently in R.H. Stote, D.J. States and M. Karplus, J. Chimie Physique (1991)]

Atom electrostatics

Atom electrostatics indicates that interactions are computed on an atom-atom pair basis. There are two options that specify the radial energy functional form. The keywords CDIE and RDIE select the basic functional form. The SWIT and SHIF keywords determine the long-range truncation option.

[ ATOM ] [ CDIElec ] [ SHIFted ] [ RDIElec ] [ SWITched ] [ FSWItch ] [ FSHIft ] [ EIPS ]

CDIE

Constant dielectric. Energy is proportional to 1/R.

RDIE

Distance dielectric. Energy is proportional to 1/(R-squared)

SWIT

Switching function used from CTONNB to CTOFNB values.

SHIF

Shifted potential acting to CTOFNB and zero beyond.

FSWI

Switching function acting on force only. Energy is integral of force.

FSHI

Classical force shift method for CDIE (force has a constant offset).

EIPS

Isotropic periodic sum method for CDIE or RDIE electrostatic energy

Group Electrostatics

electrostatics-spec::= [ GROUp [ EXTEnded [ GRADients ] [ QUADrip ] ] ] [ CDIElec ] [ SWITched ] [ [ [ NOGRad ] [ NOQUads ] ] ] [ RDIElec ] [ FSWItch ] [ [ NOEXtended ] ] [ [ EIPS ] ]

SWIT

Switching function used from CTONNB to CTOFNB values.

FSWI

Switching function, but QiQj/Rcut is added before switching. (FSWI has no effect on neutral groups).

EIPS

Isotropic Periodic Sum using CTOFNB as the local region radius.

Electrostatic Distances

electrostatic-dist::= [ CUTNB real ] [ CTEXNB real ] [ CTONNB real ] [ CTOFNB real ] [ EMAX real ] [ MINE real ] [ MAXE real ] ! SOFTVDW only

EMAX

Twice the VDW energy from which the soft core potential becomes active (if SOFT key word is employed). It has a linear form for r<rc (E(rc)=EMAX/2) : Esoftl=EMAX/2+alfa*(r-rc) unless the VDWE key word turns on the exponential form : EsoftE=EMAX+alfa*r**beta For the exponential form E(0)=EMAX, so EMAX is also the VDW energy at r=0 for the exponential form

MINE

Twice the energy from which the electrostatic attractive soft potential begins. ELEE turns on the exponential form.

MAXE

Twice the energy from which the electrostatic repulsive soft potential begins. ELEE turns on the exponential form.

The soft core potential is currently implemented only in fast energy routine, so fast option has to be used to allow it. The form of the soft potential for the electrostatics is the same as for the VDW, the defaults are different for VDW, electrostatic repulsion and attraction. The defaults need to be modified for some cases (i.e. for the spc water model) to prevent shifting of energy minima. The SOFT key word turns on both VDW and electrostatic soft core. To turn off the soft core set EMAX = 0 with the SOFT keyword (in energy, minimization or nbonds call).

CTEXNB

defines the cutoff distance beyond which interaction pairs are excluded from the Extended Electrostatics calculation.

Extended (group) Electrostatics

electrostatics-spec::= [ ATOM ] [ CDIElec ] [ SHIFted ] [ GROUp [ EXTEnded [ GRADients ] [ QUADrip ] ] ] [ RDIElec ] [ SWITched ] [ [ [ NOGRad ] [ NOQUads ] ] ] [ [ NOEXtended ] ]

EXTE

invokes the extended electrostatics command for calculating long range electrostatic interactions.

NOEX

suppress the extended calculation.

GRAD

keyword flags the inclusion of the field of the extended gradient in calculating the force on each atom,i.e. include first and second derivatives.

QUAD

flags the inclusion of the quadrupole in the multipole expansion.

Reaction Fields

misc-specs::= [ EPS real ] [ E14Factor real ] [ NORXN ] [ RXNFLD rxnfld-spec ] [ RXNNB rxnfld-spec ] rxnfld-spec::= [ EPSEXT real ] [ ORDER integer ] [ SHELL real ]

Isotropic Periodic Sum

electrostatics-spec::=[EIPS] [ ATOM ] [ CDIElec ] [ GROUp ] [ RDIElec ]

Van Der Waal Interactions

The VDW keyword (default) invokes the van der Waal energy term. To supress this term, the NOVDw keyword may be used.

Distance specified van der Waal Function

[ VGROUP [ VSWITched ] ] [ [ VIPS ] ] vdw-spec::= [ VATOM [ VSHIfted ] ] [ [ VSWItched ] ] [ [ VFSWitch ] ] [ [ VIPS ] ]

VIPS

Isotropic Periodic Sum for VDW interaction.

Miscellaneous options and keywords

Dielectric specification

misc-specs::= [ EPS real ] [ E14Factor real ]

Warning Distance Specifications

warning-dist::= [ WMIN real ] [ WRNMXD real ]

WRNMXD

keyword defines a warning cutoff for maximum atom displacement from the last close contact list update (used in EXTEnded)

Initialization

In all cases as many keywords and values as desired may be specified. The key words, their functions, and defaults are:

The method to be used

Distance cutoff in generating the list of pairs

CUTNB value (default 8.0)

Distance cut at which the switching function eliminates all contributions from a pair in calculating energies. Once specified, This value is not reset unless respecified.

CTOFNB value (default CUTNB-0.5)

Distance cut at which the smoothing function begins to reduce a pair’s contribution. This value is not used with SHFT. Once specified, This value is not reset unless respecified.

CTONNB value (default CTOFNB-1.0)

Dielectric constant for the extended electrostatics routines (RDIE option sets the dielectric equal to r times the EPS value)

EPS value (default 1.0 for r dielectric) EPS 0.0 or NOELec (zero elecrostatic energy)

Warning cutoff for minimum atom to atom distance. Pairs are checked during close contact list compilation.

WMIN value (default 1.5)

Warning cutoff for maximum atom displacement from the last close contact list update (used only in EXTEnded)

WRNMXD value (default 0.5)

The presence of soft core nonbonded interactions (turned on only if SOFT key word is present)

Exclusion Lists

By default, vdw and electrostatic interactions between two bonded 1-2 interactions) and two atoms bonded to a common atom (1-3 interactions) atoms are excluded from the calculation of energy and forces. Also, special vdw parameters and an electrostatic scale factor (E14FAC) can be applied to atom pairs separated by 3 bonds (1-4 interactions). The control of the exclusion list is by the integer variable, NBXMod.

NBXMod = 0 Preserve the existing exclusion lists NBXMod = +/- 1 Add nothing to the exclusion list NBXMod = +/- 2 Add only 1-2 (bond) interactions NBXMod = +/- 3 Add 1-2 and 1-3 (bond and angle) NBXMod = +/- 4 Add 1-2 1-3 and 1-4'S NBXMod = +/- 5 Add 1-2 1-3 and special 1-4 interactions

A positive NBXMod value causes the explicit exclusions in the PSF (inb array) to be added to the exclusion list. A negative value causes the use of only the bond connectivity data to construct the exclusion list (thus, ignoring the PSF data).

LRC

Long range correction to cutoff van der Waal’s energy and its contribution to the pressure.

Isotropic Periodic Sum (IPS)

This is a newly developed method to calculate electrostatic and/or VDW interaction accurately and efficiently. Using EIPS, VIPS to setup IPS calculation for electrostatic and vdw interactions, respectively. Both atom and group nonbonded list are supported. Alternative, one can set IPS for both electrostatic and vdw interactions. Also, one can use VIPS for vdw and use Ewald for charge interaction. The IPS method can be applied to both periodic and finit(in vacuum) systems. The IPS method calculate long-range interactions in two steps. The first step calculates the interaction with the local region defined by CTOFNB or RIPS. The second step calculates the difference of an anisotropic system (defined by radius RAIPS, which is set by default the diagonal distance of the PBC box. The first step is done the same way as the cutoff methods by summing over local atom pairs. The second step is done through the convolution theorem which can be efficiently calculated using FFT technique. This method can be used for both homogenous and heterogeneous systems. By setting PXYZ, the second step will be turned off by assuming the system is fully homogenous. Setting PXY|PYZ|PXZ or PX|PY|PZ will use 1+2D IPS for the second step calculation. Other than default values determined by system sizes, grid numbers for FFT can be set through MIPSX, MIPSY,MIPSZ, and Bspline order by MIPSO. If use VIPS with EWALD (PME), grid numbers and bspline order will be defined by PME input. For constant pressure simulation, the IPS energies at grid points are updated according to the updating frequency, NIPSFRQ (default 1), and the volume change ratio, DVBIPS(default 10e-9). Increase NIPSFRQ or DVBIPS can slightly speed up simulation.

For finite systems( such as in vacuum), the calculation is done by assuming a PBC box that is large enough (twice the size of the system) so that molecule will not see any images within the interaction range.

The original description of the IPS method can be found at: “Xiongwu Wu and Bernard R. Brooks, Isotropic Periodic Sum: A method for the calculation of long-range interactions. J. Chem. Phys., Vol.122, No.4, article 044107 (2005)” (http://link.aip.org/link/?JCP/122/044107/1)

Here are some examples to using IPS. Using IPS for any simulation system:

DYNA LEAP CPT STRT NSTE 100 TIME 0.001 - EIPS VIPS - CUTNB 12 CTOFNB 10 imgfrq 10 inbfrq 10

or use IPS for vdw and Ewald for charge interaction:

DYNA LEAP CPT STRT NSTE 100 TIME 0.001 - VIPS - ATOM EWALD PMEWALD KAPPA 0.32 FFTX 32 FFTY 32 FFTZ 32 ORDER 6 - CUTNB 12 CTOFNB 10 IMGFRQ 10 INBFRQ 10 NTRFRQ 100

or for fully homogenous systems:

DYNA LEAP CPT STRT NSTE 100 TIME 0.001 - EIPS VIPS PXYZ - CUTNB 12 CTOFNB 10 IMGFRQ 10 INBFRQ 10

or for interfacial systems using 1+2D IPS:

DYNA LEAP CPT STRT NSTE 100 TIME 0.001 - EIPS VIPS PXY - CUTNB 12 CTOFNB 10 IMGFRQ 10 INBFRQ 10

There are four algorithms used in calculating the nonbonded energies, each making different approximations in an attempt to speed the calculation. Electrostatic interactions are the most difficult to deal with for two reasons. They do not fall off quickly with distance (so it is inappropriate to simply ignore all interactions beyond some cutoff), and they depend on odd powers of r necessitating expensive square root calculations for each pair evaluated. The approximations used to make the electrostatics calculation more tractable are setting the dielectric constant equal to r or using a constant dielectric but only calculating distant interactions periodically (and storing the value in between).

Setting the dielectric constant equal to the atom atom distance times a constant factor ( determined by the EPS keyword value ) makes the computation easier by eliminating the need to calculate square roots and by making the calculated contribution fall off more quickly. It also introduces problems. The force calculated using an r dependent dielectric will be larger than the force from a constant dielectric at short distances (5.0 angstroms or less by comparison to a constant dielectric of 2.5). In addition, the electrostatic contribution still falls off relatively slowly and large distance cutoffs are needed. As the number of atom pairs included will be proportional to the cutoff cubed, this is a significant disadvantage.

The SHIFt option is similar to SWITch except, the potential:

```
E = (QI*QJ/(EPS*R)) * ( 1 - (R/CTOFNB)**2 )**2
```

is used when ( R < CTOFNB ) and zero otherwise. This potential and it first derivative approach zero as R becomes CTOFNB, without the messy computation of switching functions and steep forces at large R.

CDIE uses a constant dielectric everywhere. This requires a square root to be calculated in the inner loop of ENBOND, slowing things down a bit, but it is physically more reasonable and widely employed by other groups doing empirical energy modeling. The short range forces are identical to those calculated with the other options, reflecting the decrease in dielectric shielding at short ranges.

The constant dielectric routines compile the close contact list using the same two stage minimum rectangle box search that is described above. In this way the efficiency of a residue by residue search is exploited while being certain that all necessary pairs are included. For close residue pairs an atom by atom search is then performed. Atom pairs are either included in the list of close contacts or their electrostatic interactions are calculated and stored.

For the long range forces there is effectively no cutoff in the electrostatic energy when using the Extended Electrostatics model. The Extended Electrostatics model approximates the full electrostatic interaction by partitioning the electric potential and the resulting forces at any point ri into a near and extended contribution. The near contribution arises from the charged particles which are spatially close to ri while the extended contribution arises from the particles which are spatially distant from ri. The total electrostatic potential can be written as a sum of the two. The near region is defined in terms of a radial distance, CUTNB, for each atom. Interactions between atoms separated by a distance greater than CUTNB are calculated using a time saving multipole approximation when the nbond list is updated. These interactions are stored together with their first (NOGRad) or first and second (GRADients) derivatives. Interactions between particles within CUTNB are calculated by the conventional pairwise additive scheme. (For a more complete development of the model, see R.H. Stote, D.J. States and M. Karplus, J. Chimie Physique Vol. 11/12, 1991). The energy is calculated by explicitly evaluating pairs in the list and using the stored potentials, fields, and gradients to approximate the distant pairs. In essence the routines assume that for distant pairs the atom movements will be small enough that the changes in their electrostatic interactions can be accurately calculated using local expansions. In using this model the GROUp method for constructing the nonbond list must be used. The interactions between particles within CUTNB are truncated rather than having a SHIFt or SWITch function applied. Additionally, as one is calculating all electrostatic interactions in the system, the dielectric constant should be set to 1.0.

Not Available at this time:

An option is offered to increase the accuracy of residue residue interactions by using a multipole expansion of one residue evaluated for each atom of the other. This cutoff for this treatment is CUTMP. For residue pairs outside of CUTMP only a single multipole evaluation is made and second order polynomial expansion is used to extrapolate to each atom. Ordinarily this is sufficient and CUTMP is set to 0.0.

The initialization and list compilation is performed by the subroutine NBONDS which in turn will call a lower level routine that will do all of the work. It functions by guessing how much space will be needed to store the close contact list, allocating that space (and space for electrostatic potentials and gradients if necessary) on the heap, and calling the appropriate subroutine to actually compile the nonbonded list (NBONDG,...). If sufficient space was not available 1.5 times as much is allocated and another attempt is made.

ENBOND evaluates the nonbonded energy, calling EEXEL to evaluate the stored electric potentials and fields. Double precision is used for all arithmetic.

All of the nonbonded cutoffs and lists are stored on the heap. BNBND is the descriptor array passed through most of the program (in some of the analysis routines an additional array BNBNDC is used for the comparison data structure). BNBND holds heap addresses and LNBND holds the lengths of the elements in the data structure. To actually access the data it is necessary to include INBND.FCM (an index common block) and specify HEAP(BNBND(xxx)) where xxx is the desired element name in INBND.FCM. This is arrangement has the advantage of allowing dynamic storage allocation and easy modification of the types of information passed from routine to routine.

The nonbonded data structure is described in: source/fcm/inbnd.fcm

The option EXOForce, forces update of exclusion lists. The option SOFT will turn on the soft core nonbonded potential.

The option GRAPe will perform all the nonbond interactions in the specialized molecular dynamics hardware, called GRavity PipE. There is an environment (M2_ON) variable to specify which board to use:

Example for MDGRAPE2:

```
envi m2_on 0 (use 1st board)
envi m2_on 1 (use 2nd board)
envi m2_on "0,1" (use both: 1st and 2nd)
```

If you don’t specify this environment variable then it will use all available boards. This variable is changed for MDGRAPE3 to MR3_BOARD. grap-spec is an integer variable (default -1 = don’t use GRAPE) If zero then normal usage. NOTE for GRAPE: Because of the energy calculation is done only whe it is printed in the output, default ECHECK is too low. Please increase!

The option NOLIst will perform all the no cuttof nonbond interactions without the use of nonbond list, the same as on GRAPE machine.

There are two independent implementations of table lookups. LOOKup uses fast lookups into internally generated tables (using standard CHARMM FAST energy routines), and ETABLE reads in tables from external files.

Fast non-bonded force and energy calculation for standard MD simulations, in particular simulations in explicit solvent (water). Not working with (or not tested with) BLOCK, TSM, PERT, the various implicit solvent models. Speedup depends on the size of the system and the number of water molecules but typically a 2-fold speedup may be obtained compared to the standard fast expanded routines.

Reference: Nilsson, L. “Fast lookup tables for pairwise interactions” (2007), in preparation.

Non-bonded interactions within a user specified selection of solvent molecules (any three site water model with the O first, and followed by two identical hydrogens in the PSF should work) are removed from the regular non-bonded lists and are instead handled by a dedicated routine. Speedup is achieved through the use of a table lookup of energies and forces. It is easy to extend to other solvent models, although a table lookup may be inefficient if there are more than a few atomtypes in the model.

Similar tables are used for interactions between the selected solvent molecules and the rest of the system (“solvent-solute”) and for the solute-solute interactions, except 1-4 interactions which are sent on to the standard routine.

The code works with PBC (images/crystal) and in parallel, with all cutoff methods implemented in the ENBAEXP routine (eg, FSHIFT,FSWITCH,VSHIFT and the real space part of PME).

Works with nonbond-list methods that generate an atom based non-bond list (BYGRoup, BYCBim).

No second derivatives.

Use:

```
LOOKup { RESEt }
{ atom-selection [[NO]INTerpolate] [TABIncr <int>] [[NO]ENERgy] -
[NOVU] [NOUU] }
RESEt The regular routines will be used
INTerpolate Linear interpolation will be used in the table lookup
NOINterpolate
TABIncrement Determines resolution and size (=TABINCREMENT*CTOFNB**2) of
lookup tables, which are indexed using Rij**2. For instance
the energy of two water oxygens at a distance of R is found
in EOO(R*R*TABINCREMENT).
NOENergy Energies will only be evaluated when non-bond list is updated
This may result in apparent ECHEck violations; if so you can
increase ECHEck (or turn off the check: ECHEck -1.0)
ENERgy Energies will always be evaluated
NOVU Do not use lookup fopr the solVent-solUte interactions
NOUU Do not use lookup fopr the solUte-solUte interactions.
NB! 1-4 interactions are always handled by standard routine
atom-selection Select the waters to be included. This is mandatory.
DEFAULTS:
NOENergy; INTErpolation; TABIncrement 20; no selection; use solvent-solute
and solute-solute lookup routines.
```

Increasing TABI, using the ENERGY and INTERPOLATE options will increase the size of the lookup tables, which may reduce the speed of the calculation if it leads to cache misses - experimentation is advised. With INTERpolate you can use a smaller TABI and get good accuracy.

Everything is reset on each invocation.

Usage example (see also test/c34test/lookup.inp):

```
! When the system is completely defined (PSF,cooordinateds, IMAGES/CRYSTAL,,,)
! first a call to energy to fill interaction coefficient arrays
ENERGY FSHIFT VSHIFT CDIE CUTNB 14.0 CTOFNB 12.0
LOOKUP SELE SEGI WAT END
ENERGY / MINIMIZE / DYNAMICS - but do NOT change the cutoff distances
or options!
```

Implementation:

Code is protected by ##LOOKUP pref.dat keyword

LOOKUP | Identifies the water molecules to get special treatment (or turns the whole thing off). Fills the lookup tables using calls to ENBAEXP - so it is important that an energy evaluation has already been performed in order to have all coefficient arrays properly filled. When the non-bond list has been generated in NBONDS, all atom pairs belonging to this set are removed from the regular and image lists, and placed in their own lists (one for water-water oxygen-oxygen pairs, and one each for solvent-solute and solute-solute (non 1-4) atom pairs). This scheme should work transparently for all methods as long as the atom based lists are generated in the first place. There is a slight penalty for this second pass through the list, but the implementation is very non-intrusive. Routine ELOOKUP is called from ENERGY (and EIMAGE) to compute the water-water and water-solute interactions as specified. Since there is only one table for solvent-solvent energies all the solvent-solvent nonbond energy (vdW+Coulomb) computed in EWWNB is reported as electrostatic, with zero returned for EvdW; the solvent-solute part separates Coulomb and vdW. Specifying NOENenergy skips the energy lookups, except at nonbond list updates. This can give a modest (5-10%) speedup, at the risk of getting caught by ECHECK; ECHECK can be turned off by ECHECK -1.0. The tables are stored in single precision and some intermediate variables are also in single precision. For typical systems the relative error in total energy and GRMS is of the order of 10^-4. Using the INTErpolation option and/or increasing TABIncrement can reduce this somewhat, but the error seems to be dominated by the single precision noise rather quickly. |

The nonbond energy terms may also be specified with a user supplied binary lookup table. The command

`READ TABLE UNIT int`

will invoke this feature and disable all other energy term options. The nonbond list specifiers will still be used (cutoff distances...).

This is completely separate from the NBSOlv method outlined above.

This feature is not designed for casual users, and is not supported with test cases. Also, in version17, there is an uncorrected bug in the second derivative determination.

To use this feature, first read the common file ETABLE.FCM for a description of variables, and then create a file the the routine REATBL (consult the source) can read. The sources for this option are contained in the file ETABLE.FLX.

The purpose of using finite cutoffs in energy calculations is to reduce, from O(N*N) to O(N), the number of nonbonded interaction terms that actually need to be computed. CHARMM has three ways of building the nonbonded interaction list: BYGRoups, BYCUbes, and ‘By-Clusters-in-Cubes,’ or BYCC. The BYCC method is a combination of the earlier BYGR and BYCU methods. For a given set of atoms and a given cut-off distance, all three algorithms should generate the same non-bonded list.

BYCBim extends the BYCUbes method to systems with images or periodic boundaries, but is more restricted in other options that are compatible with it. It will not work with replica, extended electrostatics, or constant pressure dynamics since no group list is generated. However, BYCBim does work in parallel mode; BYCUbes does not.

The differences between BYCUbes, BYGRoups, and BYCC are in speed and memory requirements. This section gives a description of the algorithms, followed by a description of some of the unique aspects of BYCC. Finally, a few general guidelines for choosing between the earlier BYCUbes and BYGRoups methods are presented.

The basis of the efficiency of the BYGRoups algorithm over a brute-force comparison of each atomic position with all others in the system is that it clusters atoms into chemical groups, initially ignoring the individual atoms, themselves. This significantly reduces the number of pairs of particles that need to be examined. Effectively, BYGR speeds up the calculation by reducing the particle density, which it does by simply redefining the particles. Once a list of group-group pairs satisfying the initial distance criterion is made, only atoms from this relatively short list are then considered for further atom- atom distance testing.

In contrast, the efficiency of the BYCUbes algorithm is based on the partitioning of the system into small cubical regions. A synopsis of how it works:

- Find a rectangular parallelepiped that bounds the system (with margins), is aligned with the Cartesian axes, and has sides that are integral multiples of the cutoff distance. Divide this parallelepiped, or box, into cubes whose sides are equal in length to the cutoff distance.
- Identify the cube that contains each atom.
- For each cube C, loop over each atom A contained in C. For each such A, loop over each atom A’ contained in the 27-cube surrounding region, which is the (3 cube x 3 cube x 3 cube) region that contains the central cube. If the A–A’ pair falls within the cutoff distance, check various other inclusion criteria; e.g. that the pair is not a 1-2 or 1-3 excluded pair.

Hence, BYCU speeds up the generation of the non-bonded list by distance- testing only pairs of atoms that are in nearby regions.

The ‘By-Clusters-in-Cubes’, or BYCC, algorithm incorporates both the partitioning technique of BYCUbes and also the atomic grouping technique of BYGRoups. BYCC divides the system into a cubical grid and compares particles only in adjacent cubes, as BYCU does. However, the particles it compares are clusters of atoms, in the spirit of BYGR, as opposed to individual atoms. The use of both techniques in BYCC allows for greater efficiency than is possible using either technique alone, and for this reason BYCC is generally faster than the other algorithms. In addition, because it does the final atom-atom distance calculations, the exclusions, and the formation of the non-bonded list all in one final loop, the routine needs to store only a cluster-cluster pair- list as a work array. Thus, the memory requirements are reduced relative to BYCU, which stores a much longer atom-atom pairlist (essentially a second non-bonded list) internally.

Like BYCU, the computational time for BYCC increases linearly with the number of atoms and sigmoidally as a function of cut-off distance. When (cut-off distance) << (radius of system) the time dependence on cut-off distance is essentially third order, but depends on the system, the machine type, and the actual cutoff value. The computational time for BYGR increases quadratically with the size of the system. NB: The energy calculations are always O(N) if a non- bonded list is being used, regardless of which method is used to generate the list. However, generation of the list itself can be either O(N*N) (e.g. BYGR) or O(N) (e.g. BYCU or BYCC).

The BYCC option, like the others, supports calculations based on chemical groups instead of atoms, and an additional option for extended electrostatics. However, clusters are not used in group-based calculations, since their role is in effect subsumed by the groups. The relative speed advantage of BYCC over BYCU is therefore diminished in group-based computations.

For the BYCC or BYCU options, when extended electrostatics are requested, CPU time for nonbonded list generation and calculation of extended electrostatics will depend on the extended electrostatics cut-off distance, CTEXNB. Hence, smaller CTEXNB values will significantly speed up the calculations. For BYGR, CPU time is independent of CTEXNB.

Note that for group-based calculations with either BYCC, BYCU, or BYGR, the groups are treated as point-like. This means that, unlike the case for cluster size in atom-based calculations, there is no dependence of CPU time on group size in the generation of the non-bonded list.

With the BYCC option, a requirement for the atom-based calculations is that clusters of atoms need to be created. A cluster is a set of atoms that have mutually close connectivity relationships. Generally, small, dense, uniformly-sized clusters yield the most efficient non-bonded list generation. The use of a separate clustering scheme instead of the usual chemical grouping arrangement used elsewhere in CHARMM (e.g. in the BYGRoups algorithm) provides a separate spatial organization of the system that can be manipulated and optimized largely independently of the other CHARMM functions. Because the criteria for the grouping of atoms are chemical, whereas those for the clustering of atoms are spatial, the optimal arrangement of atoms for CHARMM groups is not necessarily (and, indeed, not usually) the optimal arrangement for clusters.

Clusters are generated by default just before the non-bonded list is updated for the first time with BYCC invoked. Thereafter, they are regenerated when a change in the connectivity or the topology of the system is detected.

Clusters can also be (re)made with the command:

```
MKCLuster [CADL] [CMARgin] [EXCL]
```

This command is no longer required for the use of BYCC (since its equivalent is called by default), but can be used in the middle of a CHARMM run (for example to change the CADL or CMAR parameters) or to ensure reformation of atom clusters.

Keyword CADL

Consecutive atom distance limit – distance between consecutive atoms (as defined by order in RTF/PSF) in a cluster beyond which the cluster will be split. Helps prevent the generation of overly large clusters. Default : 4.0 Angstroms.

Keyword CMAR Limiting cluster margin width. The cluster margin is the calculated distance that is added to the non-bonded cutoff distance, when partitioning the clusters in the system, to ensure that all atom-atom pairs are in fact included in the distance testing. This margin depends on the largest cluster size and will affect CPU time (larger margin means slower non-bonded list generation). CMAR sets the upper limit for this added margin. Default: 5.0 Angstroms. Note that while the calculated cluster margin is the width that GUARANTEES all-atom testing, in practice CMAR can often be set below the calculated value (speeding up the calculations) without changing the results. This is particularly true when there is a large spread in the cluster sizes or when there are statistical outliers. If CMAR < calculated cluster margin, a warning is issued.

Keyword EXCL The MKCL command will result in a re-generation of the exclusion table (from the current exclusion list) if the “EXCLusions” keyword is specified. This may be necessary if the connectivity of the system is altered during a CHARMM run or if the exclusion list otherwise changes.

An additional feature of the BYCC algorithm is its ability to handle “active” atom selections, which are specified with the command

`NBACtive [atom selection]`

The purpose of this feature is to allow the user to focus calculations on regions of interest without having to alter the psf or coordinate files. It allows the non-bonded list generation routine (BYCC) to ignore completely atoms that are not defined as active, so that only active atoms appear in the non-bonded list. This speeds up non-bonded updates and energy calculations, and, in addition, it allows for more selectivity in energy calculations– in dynamics and minimization, particularly.

An example of the ideal use of this command is the study of a single subunit in a protein containing several. Another is the study of a single side chain, or a group of side chains, on a fixed protein backbone.

If the energy calculations on the active portion of the system are to be “consistent” with the inactive portion of the system, a buffer region (ideally of width CUTNB) is required surrounding the region of interest. This buffer region should be active and fixed.

Active clusters and groups are defined automatically on the basis of the active atom selection. If no active atom selection is given, it is assumed that all atoms are active.

In its current implementation, the active atom selection does not affect the bonded energies (note to developers: this should be rectified in the future). This has at least two implications: 1) It is currently recommended that in conjunction with this command, ‘inactive’ atoms be fixed with the CONS FIX command, since otherwise they will contribute to the bonded terms but not to the non-bonded terms. 2) The connectivity between inactive and active regions, if it exists in the original system, is not broken by the NBACtive command. This means that regions defined as active must remain ‘tethered’ to the inactive regions.

The additional commands in a typical CHARMM script using the BYCC option for non-bonded list generation would be as follows: The BYCC keyword in the NBOND command is necessary for using the BYCC option.

The other commands are optional.

```
! make clusters
MKCL !no longer necessary
! define active atom set and buffer subset (optional)
DEFINE ACTIVE SELE (active atom selection) END
DEFINE BUFFER SELE (buffer atom selection) END !subset of ACTIVE
! "activate" the selected atoms (optional)
NBACtive SELE ACTIVE END !default is all atoms
! fix inactive atoms (optional)
CONS FIX SELE ((.NOT.ACTIVE) .OR. BUFFER) END
! call non-bonded list generation routine
NBONd -
BYCC !necessary
```

Given any pair of functions F1(N) and F2(N), which are O(N) and O(N*N) respectively, there will always be some constant N0 such that F2(N) > F1(N) for all N > N0. This constant N0 may be referred to as the “payoff threshold,” since it is the system size above which the BYCUbes algorithm will be faster than BYGRoups for a given cut-off distance, particle density, machine type, and set of options.

The following are some properties of the payoff threshold that can be used as guidelines for choosing between BYGRoups and BYCUbes:

- The payoff threshold is smaller on a vector machine than on a scalar machine. That is, BYCUbes is more vectorizable than BYGRoups.
- The shape of the system does not have a big impact on the payoff threshold. BYCUbes operates by drawing a rectangular box around the system and dividing it into cubes, but on a serial machine the empty cubes take little time.
- The payoff threshold usually grows steeply with cutoff distance. This is because the speed advantage of the BYCUbes algorithm is based upon the compartmentalization of the system. The smaller the compartments (cubes), the more advantage. Since the compartment size is based on the cutoff distance, as the cutoff distance increases, there is less advantage for BYCUbes over BYGRoups.
- BYCUbes trades memory for time. Its memory requirements are significantly higher than those of BYGR and they may be prohibitive for large systems.