SklogWiki - User contributions [en]

Hard cube model

2013-04-29T11:49:40Z

Nice and Tidy: Removed stub template

[[Image:cube.png|thumb|right]]
The '''Hard cube model''' models cube-shaped particles interacting purely through excluded-volume interactions. The phase behavior has been studied extensively
<ref>[http://dx.doi.org/10.1038/nmat2959 Umang Agarwal and Fernando A. Escobedo "Mesophase behaviour of polyhedral particles", Nature Materials '''10''' pp. 230-235 (2011)]</ref>
<ref>[http://dx.doi.org/10.1073/pnas.1211784109 F. Smallenburg, L. Filion, M. Marechal, and M. Dijkstra "Vacancy-stabilized crystalline order in hard cubes", Proc. Natl. Acad. Sci. USA '''109''' pp. 17886-17891 (2012)]</ref>. The simple cubic crystal phase has been shown to contain a high number of mobile, delocalized vacancies, similar to those seen in [[Parallel hard cubes]].

==Implementation==
Overlaps between cubes can be checked based on the [http://en.wikipedia.org/wiki/Hyperplane_separation_theorem separating axis theorem], which says that if two convex objects are not interpenetrating, there must be an axis for which the projections of the two objects will not overlap. In the case of two
convex polyhedral particles, only a finite number of possible separating axes need to be
checked: in that case, the possible separating axes are either parallel to a normal of one
of the faces of either of the two particles, or perpendicular to the plane spanned by one
of the edges of the first particle and one of the edges of the second particle.

For two cubes, the overlap check therefore corresponds to the projection of both shapes onto 15 axes: the three edge vectors of both particles (6 in total), and the axes given by the 9 possible cross products of an edge vector of one cube with an edge vector of the other cube. The particles overlap if and only if the projections of the two particles on all 15 axes overlap as well.

==References==
<references/>

'''Related reading'''
*[http://dx.doi.org/10.1063/1.4734021 Umang Agarwal and Fernando A. Escobedo "Effect of quenched size polydispersity on the ordering transitions of hard polyhedral particles", Journal of Chemical Physics '''137''' 024905 (2012)]
[[category: models]]

Janus particles

2013-04-29T11:49:04Z

Nice and Tidy: Removed stub template

[[Image:Janus.png|thumb|right| Artists impression of a Janus particle]]
'''Janus particles''' are particles consisting of (at least) two parts with different interactions. For example, the typical Janus particle is a sphere which has a short-range attraction on one half of the particle, but is purely repulsive on the other side. However, the term has also been used for non-spherical particles. The name derives from the two-faced Roman god Janus. Janus particles can be considered as a one-patch [[Patchy particles|patchy particle]].

Experimentally, the different interactions can be achieved by (for example) making the two parts of the surface hydrophobic and hydrophilic, positively and negatively charged, or smooth and rough (leading to different interactions in the presence of [[Depletion force|depletants]]). In simulations, these particles are often modeled using the [[Kern and Frenkel patchy model|Kern-Frenkel]] interaction potential.

==Phase diagram==
<ref>[http://dx.doi.org/10.1103/PhysRevLett.103.237801 Francesco Sciortino, Achille Giacometti, and Giorgio Pastore "Phase Diagram of Janus Particles", Physical Review Letters '''103''' 237801 (2009)]</ref>

==See also==
*[[Dipolar Janus particles]]
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.3177238 Steve Granick, Shan Jiang, and Qian Chen "Janus particles", Physics Today '''62''' pp. 68-69 (July 2009)]
*[http://dx.doi.org/10.1063/1.4707954 Gerald Rosenthal, Keith E. Gubbins, and Sabine H. L. Klapp "Self-assembly of model amphiphilic Janus particles", Journal of Chemical Physics '''136''' 174901 (2012)]
*[http://dx.doi.org/10.1063/1.4793626 Miguel Ángel G. Maestre, Riccardo Fantoni, Achille Giacometti, and Andrés Santos "Janus fluid with fixed patch orientations: Theory and simulations", Journal of Chemical Physics '''138''' 094904 (2013)]

[[category: phase diagrams]]

Structure factor

2011-09-16T07:56:09Z

Nice and Tidy: Slight tidy

The '''structure factor''', <math>S(k)</math>, for a monatomic system is defined by:

:<math>S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~dr</math>

where <math>k</math> is the scattering wave-vector modulus

:<math>k= |\mathbf{k}|= \frac{4 \pi }{\lambda \sin \left( \frac{\theta}{2}\right)}</math>

The structure factor is basically a [[Fourier analysis | Fourier transform]] of the [[pair distribution function]] <math>{\rm g}(r)</math>,

:<math>S(|\mathbf{k}|)= 1 + \rho \int \exp (i\mathbf{k}\cdot \mathbf{r}) \mathrm{g}(r) ~\mathrm{d}\mathbf{r}</math>

At zero wavenumber, ''i.e.'' <math>|\mathbf{k}|=0</math>,

:<math>S(0) = k_BT \left. \frac{\partial \rho}{\partial p}\right\vert_T</math>

from which one can calculate the [[Compressibility | isothermal compressibility]].

To calculate <math>S(k)</math> in [[Computer simulation techniques |molecular simulations]] one typically uses:

:<math>S(k) = \frac{1}{N} \sum^{N}_{n,m=1} \langle\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m)) \rangle </math>,

where <math>N</math> is the number of particles and <math>\mathbf{r}_n</math> and
<math>\mathbf{r}_m</math> are the coordinates of particles
<math>n</math> and <math>m</math> respectively.

The dynamic, time dependent structure factor is defined as follows:
:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} \langle \exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0))) \rangle </math>,

The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known
as the collective (or coherent) intermediate scattering function.
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, '''6''' pp. 8415-8427 (1994)]
[[category: Statistical mechanics]]

Phase diagrams: Pressure-temperature plane

2011-06-16T12:33:49Z

Nice and Tidy: Slight tidy

{{Stub-general}}
The following is a schematic [[Phase diagrams |phase diagram]] of a monatomic substance in the [[temperature]]-[[pressure]] plane.
It shows the vapour and liquid phases, as well as the crystalline solid phase.
The [[critical points |critical point]] is highlighted by a red spot, and the green spot represents the
[[triple point]].
[[Image:press_temp.png|center|450px]]
==See also==
*[[Density-temperature]]
*[[Binary phase diagrams]]

[[category:phase diagrams]]

Exact solution of the Percus Yevick integral equation for hard spheres

2011-04-05T09:56:48Z

Nice and Tidy: Sligh ttidy + Cite format of references

The exact solution for the [[Percus Yevick]] [[Integral equations |integral equation]] for the [[hard sphere model]]
was derived by M. S. Wertheim in 1963 <ref name="wertheim1" >[http://dx.doi.org/10.1103/PhysRevLett.10.321 M. S. Wertheim "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres", Physical Review Letters '''10''' 321 - 323 (1963)]</ref> (see also <ref>[http://dx.doi.org/10.1063/1.1704158 M. S. Wertheim "Analytic Solution of the Percus-Yevick Equation", Journal of Mathematical Physics, '''5''' pp. 643-651 (1964)]</ref>), and for [[mixtures]] by Joel Lebowitz in 1964 <ref>[http://dx.doi.org/10.1103/PhysRev.133.A895 J. L. Lebowitz, "Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres", Physical Review '''133''' pp. A895 - A899 (1964)]</ref>.

The [[direct correlation function]] is given by (Eq. 6 of <ref name="wertheim1" > </ref> )

:<math>C(r/R) = - \frac{(1+2\eta)^2 - 6\eta(1+ \frac{1}{2} \eta)^2(r/R) + \eta(1+2\eta)^2\frac{(r/R)^3}{2}}{(1-\eta)^4}</math>

where

:<math>\eta = \frac{1}{6} \pi R^3 \rho</math>

and <math>R</math> is the hard sphere diameter.
The [[Equations of state | equation of state]] is given by (Eq. 7 of <ref name="wertheim1" > </ref>)

:<math>\frac{\beta P}{\rho} = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math>

where <math>\beta</math> is the [[inverse temperature]]. Everett Thiele also studied this system <ref>[http://dx.doi.org/10.1063/1.1734272 Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics, '''39''' pp. 474-479 (1963)]</ref>,
resulting in (Eq. 23)

:<math>\left.h_0(r)\right. = ar+ br^2 + cr^4</math>

where (Eq. 24)

:<math>a = \frac{(2x+1)^2}{(x-1)^4}</math>

and

:<math>b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}</math>

and

:<math>c= \frac{x(2x+1)^2}{2(x-1)^4}</math>

and where <math>x=\rho/4</math>.

The [[pressure]] via the pressure route (Eq.s 32 and 33) is

:<math>P=nk_BT\frac{(1+2x+3x^2)}{(1-x)^2}</math>

and the [[Compressibility equation |compressibility]] route is

:<math>P=nk_BT\frac{(1+x+x^2)}{(1-x)^3}</math>

==References==
<references/>

[[Category: Integral equations]]

Martyna-Tuckerman-Tobias-Klein barostat

2011-02-19T12:24:08Z

Nice and Tidy: Slight tidy.

{{Stub-general}}
'''Martyna-Tuckerman-Klein barostat''' <ref>[http://dx.doi.org/10.1080/00268979600100761 G. J. Martyna, M. E. Tuckerman, D. J. Tobias and M. L. Klein "Explicit reversible integrators for extended systems dynamics", Molecular Physics '''87''' pp. 1117-1157 (1996)]</ref>.
==References==
<references/>
[[category: molecular dynamics]]

Hard spherocylinders

2010-11-30T16:43:12Z

Nice and Tidy:

[[Image:spherocylinder_purple.png|thumb|right]]
The '''hard spherocylinder''' model consists of an impenetrable cylinder, capped at both ends
by hemispheres whose diameters are the same as the diameter of the cylinder. The hard spherocylinder model
has been studied extensively because of its propensity to form both [[Nematic phase | nematic]] and [[Smectic phases | smectic]] [[Liquid crystals | liquid crystalline]] phases <ref>[http://dx.doi.org/10.1038/332822a0 D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants "Thermodynamic stability of a smectic phase in a system of hard rods", Nature '''332''' p. 822 (1988)]</ref>. One of the first studies of hard spherocylinders was undertaken by Cotter and Martire <ref>[http://dx.doi.org/10.1063/1.1673232 Martha A. Cotter and Daniel E. Martire "Statistical Mechanics of Rodlike Particles. II. A Scaled Particle Investigation of the Aligned to Isotropic Transition in a Fluid of Rigid Spherocylinders", Journal of Chemical Physics '''52''' pp. 1909-1919 (1970)]</ref> using [[scaled-particle theory]], and one of the first simulations was in the classic work of Jacques Vieillard-Baron <ref>[http://dx.doi.org/10.1080/00268977400102161 Jacques Vieillard-Baron "The equation of state of a system of hard spherocylinders", Molecular Physics '''28''' pp. 809-818 (1974)]</ref>. In the limit of zero diameter the hard spherocylinder becomes a line segment, often known as the [[3-dimensional hard rods |hard rod model]].
==Volume==
The molecular volume of the spherocylinder is given by

:<math>v_0 = \pi \left( \frac{LD^2}{4} + \frac{D^3}{6} \right)</math>

where <math>L</math> is the length of the cylindrical part of the spherocylinder and <math>D</math> is the diameter.
==Minimum distance==
The minimum distance between two spherocylinders can be calculated using an algorithm published by Vega and Lago <ref>[http://dx.doi.org/10.1016/0097-8485(94)80023-5 Carlos Vega and Santiago Lago "A fast algorithm to evaluate the shortest distance between rods", Computers & Chemistry '''18''' pp. 55-59 (1994)]</ref>. The [[Source code for the minimum distance between two rods | source code can be found here]]. Such an algorithm is essential in, for example, a [[Monte Carlo]] simulation, in order to check for overlaps between two sites.
==Virial coefficients==
:''Main article: [[Hard spherocylinders: virial coefficients]]''
==Phase diagram==
:''Main aritcle: [[Phase diagram of the hard spherocylinder model]]''
==See also==
*[[Charged hard spherocylinders]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1021/j100303a008 Daan Frenkel "Onsager's spherocylinders revisited", Journal of Physical Chemistry '''91''' pp. 4912-4916 (1987)]
[[Category: Models]]

Hard spherocylinders

2010-11-30T16:37:40Z

Nice and Tidy: Changed references to Cite format

[[Image:spherocylinder_purple.png|thumb|right]]
The '''hard spherocylinder''' model consists of an impenetrable cylinder, capped at both ends
by hemispheres whose diameters are the same as the diameter of the cylinder. The hard spherocylinder model
has been studied extensively because of its propensity to form both [[Nematic phase | nematic]] and [[Smectic phases | smectic]] [[Liquid crystals | liquid crystalline]] phases. One of the first studies of hard spherocylinders was undertaken by Cotter and Martire <ref>[http://dx.doi.org/10.1063/1.1673232 Martha A. Cotter and Daniel E. Martire "Statistical Mechanics of Rodlike Particles. II. A Scaled Particle Investigation of the Aligned to Isotropic Transition in a Fluid of Rigid Spherocylinders", Journal of Chemical Physics '''52''' pp. 1909-1919 (1970)]</ref> using [[scaled-particle theory]], and one of the first simulations was in the classic work of Jacques Vieillard-Baron <ref>[http://dx.doi.org/10.1080/00268977400102161 Jacques Vieillard-Baron "The equation of state of a system of hard spherocylinders", Molecular Physics '''28''' pp. 809-818 (1974)]</ref>. In the limit of zero diameter the hard spherocylinder becomes a line segment, often known as the [[3-dimensional hard rods |hard rod model]].
==Volume==
The molecular volume of the spherocylinder is given by

:<math>v_0 = \pi \left( \frac{LD^2}{4} + \frac{D^3}{6} \right)</math>

where <math>L</math> is the length of the cylindrical part of the spherocylinder and <math>D</math> is the diameter.
==Minimum distance==
The minimum distance between two spherocylinders can be calculated using an algorithm published by Vega and Lago <ref>[http://dx.doi.org/10.1016/0097-8485(94)80023-5 Carlos Vega and Santiago Lago "A fast algorithm to evaluate the shortest distance between rods", Computers & Chemistry '''18''' pp. 55-59 (1994)]</ref>. The [[Source code for the minimum distance between two rods | source code can be found here]]. Such an algorithm is essential in, for example, a [[Monte Carlo]] simulation, in order to check for overlaps between two sites.
==Virial coefficients==
:''Main article: [[Hard spherocylinders: virial coefficients]]''
==Phase diagram==
:''Main aritcle: [[Phase diagram of the hard spherocylinder model]]''
==See also==
*[[Charged hard spherocylinders]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1021/j100303a008 Daan Frenkel "Onsager's spherocylinders revisited", Journal of Physical Chemistry '''91''' pp. 4912-4916 (1987)]
*[http://dx.doi.org/10.1038/332822a0 D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants "Thermodynamic stability of a smectic phase in a system of hard rods", Nature '''332''' p. 822 (1988)]
[[Category: Models]]

Time step

2010-11-26T09:22:57Z

Nice and Tidy: Added a couple of internal links

{{Stub-general}}
The '''time-step''' (often written as <math>\delta t</math>) is an important variable in [[molecular dynamics]] simulations. It is usually of the order of femto (<math>10^{-15}</math>) seconds for simulations of [[flexible molecules]].
==Multiple time steps==
<ref>[http://dx.doi.org/10.1080/00268977800100471 W. B. Streett, D. J. Tildesley and G. Saville "Multiple time-step methods in molecular dynamics", Molecular Physics '''35''' pp. 639-648 (1978)]</ref>
====RESPA====
A well known multiple time step method is the reversible reference system propagator algorithm ([[RESPA]]) <ref>[http://dx.doi.org/10.1063/1.463137 M. Tuckerman, B. J. Berne and G. J. Martyna "Reversible multiple time scale molecular dynamics", Journal of Chemical Physics '''97''' pp. 1990-2001 (1992)]</ref>.
==See also==
*[[Dissipative particle dynamics]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1063/1.3493459 P. Faccioli "Molecular dynamics at low time resolution", Journal of Chemical Physics '''133''' 164106 (2010)]
[[category:molecular dynamics]]

Velocity Verlet algorithm

2010-11-26T09:20:31Z

Nice and Tidy: Changed references to Cite format and added internal link

The '''Velocity Verlet algorithm''', for use in [[molecular dynamics]], is given by <ref>[http://dx.doi.org/10.1103/PhysRev.159.98 Loup Verlet "Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules", Physical Review '''159''' pp. 98-103 (1967)]</ref>:

:<math>r(t + \delta t) = r (t) + \delta t v(t) + \frac{1}{2} \delta t^2 a(t)</math>

:<math>v \left(t+ \delta t\right) = v(t) + \frac{1}{2} \delta t [ a(t) + a(t+\delta t)]</math>

where <math>r</math> is the position, <math>v</math> is the velocity, <math>a</math> is the acceleration and <math>t</math> is the time. <math>\delta t</math> is known as the [[time step]].
==See also==
*[[Verlet leap-frog algorithm]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1063/1.442716 William C. Swope, Hans C. Andersen, Peter H. Berens and Kent R. Wilson "A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters", Journal of Chemical Physics '''76''' pp. 637-649 (1982)]
==External resources==
*[ftp://ftp.dl.ac.uk/ccp5/ALLEN_TILDESLEY/F.04 Velocity version of Verlet algorithm ] sample FORTRAN computer code from the book [http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)].
[[category: Molecular dynamics]]

Exp6 potential

2010-11-25T11:17:11Z

Nice and Tidy: Added a redirect page

#REDIRECT[[Exp-6 potential]]

FORTRAN code for the Kolafa and Nezbeda equation of state

2010-11-25T11:13:59Z

Nice and Tidy: Added a comment re MAIN, and added spacers for column 6.

The following are the FORTRAN source code functions (without a "main" program) for the [[Lennard-Jones_equation_of_state#Kolafa and Nezbeda equation of state | Kolafa and Nezbeda equation of state]] for the [[Lennard-Jones model]]

c===================================================================
C Package supplying the thermodynamic properties of the
C LENNARD-JONES fluid
c
c J. Kolafa, I. Nezbeda, Fluid Phase Equil. 100 (1994), 1
c
c ALJ(T,rho)...Helmholtz free energy (including the ideal term)
c PLJ(T,rho)...Pressure
c ULJ(T,rho)...Internal energy
c===================================================================
DOUBLE PRECISION FUNCTION ALJ(T,rho)
C Helmholtz free energy (including the ideal term)
c
implicit double precision (a-h,o-z)
data pi /3.141592654d0/
eta = PI/6.*rho * (dC(T))**3
ALJ = (dlog(rho)+betaAHS(eta)
+ +rho*BC(T)/exp(gammaBH(T)*rho**2))*T
+ +DALJ(T,rho)
RETURN
END
C/* Helmholtz free energy (without ideal term) */
DOUBLE PRECISION FUNCTION ALJres(T,rho)
implicit double precision (a-h,o-z)
data pi /3.141592654d0/
eta = PI/6. *rho*(dC(T))**3
ALJres = (betaAHS(eta)
+ +rho*BC(T)/exp(gammaBH(T)*Sqrt(rho)))*T
+ +DALJ(T,rho)
RETURN
END
C/* pressure */
DOUBLE PRECISION FUNCTION PLJ(T,rho)
implicit double precision (a-h,o-z)
data pi /3.141592654d0/
eta=PI/6. *rho*(dC(T))**3
sum=((2.01546797*2+rho*(
+ (-28.17881636)*3+rho*(
+ 28.28313847*4+rho*
+ (-10.42402873)*5)))
+ +(-19.58371655*2+rho*(
+ +75.62340289*3+rho*(
+ (-120.70586598)*4+rho*(
+ +93.92740328*5+rho*
+ (-27.37737354)*6))))/dsqrt(T)
+ + ((29.34470520*2+rho*(
+ (-112.35356937)*3+rho*(
+ +170.64908980*4+rho*(
+ (-123.06669187)*5+rho*
+ 34.42288969*6))))+
+ ((-13.37031968)*2+rho*(
+ 65.38059570*3+rho*(
+ (-115.09233113)*4+rho*(
+ 88.91973082*5+rho*
+ (-25.62099890)*6))))/T)/T)*rho**2
PLJ = ((zHS(eta)
+ + BC(T)/exp(gammaBH(T)*rho**2)
+ *rho*(1-2*gammaBH(T)*rho**2))*T
+ +sum )*rho
RETURN
END
C/* internal energy */
DOUBLE PRECISION FUNCTION ULJ( T, rho)
implicit double precision (a-h,o-z)
data pi /3.141592654d0/
dBHdT=dCdT(T)
dB2BHdT=BCdT(T)
d=dC(T)
eta=PI/6. *rho*d**3
sum= ((2.01546797+rho*(
+ (-28.17881636)+rho*(
+ +28.28313847+rho*
+ (-10.42402873))))
+ + (-19.58371655*1.5+rho*(
+ 75.62340289*1.5+rho*(
+ (-120.70586598)*1.5+rho*(
+ 93.92740328*1.5+rho*
+ (-27.37737354)*1.5))))/dsqrt(T)
+ + ((29.34470520*2+rho*(
+ -112.35356937*2+rho*(
+ 170.64908980*2+rho*(
+ -123.06669187*2+rho*
+ 34.42288969*2)))) +
+ (-13.37031968*3+rho*(
+ 65.38059570*3+rho*(
+ -115.09233113*3+rho*(
+ 88.91973082*3+rho*
+ (-25.62099890)*3))))/T)/T) *rho*rho
ULJ = 3*(zHS(eta)-1)*dBHdT/d
+ +rho*dB2BHdT/exp(gammaBH(T)*rho**2) +sum
RETURN
END
DOUBLE PRECISION FUNCTION zHS(eta)
implicit double precision (a-h,o-z)
zHS = (1+eta*(1+eta*(1-eta/1.5*(1+eta)))) / (1-eta)**3
RETURN
END
DOUBLE PRECISION FUNCTION betaAHS( eta )
implicit double precision (a-h,o-z)
betaAHS = dlog(1-eta)/0.6
+ + eta*( (4.0/6*eta-33.0/6)*eta+34.0/6 ) /(1.-eta)**2
RETURN
END
C /* hBH diameter */
DOUBLE PRECISION FUNCTION dLJ(T)
implicit double precision (a-h,o-z)
DOUBLE PRECISION IST
isT=1/dsqrt(T)
dLJ = ((( 0.011117524191338 *isT-0.076383859168060)
+ *isT)*isT+0.000693129033539)/isT+1.080142247540047
+ +0.127841935018828*dlog(isT)
RETURN
END
DOUBLE PRECISION FUNCTION dC(T)
implicit double precision (a-h,o-z)
sT=dsqrt(T)
dC = -0.063920968*dlog(T)+0.011117524/T
+ -0.076383859/sT+1.080142248+0.000693129*sT
RETURN
END
DOUBLE PRECISION FUNCTION dCdT( T)
implicit double precision (a-h,o-z)
sT=dsqrt(T)
dCdT = 0.063920968*T+0.011117524+(-0.5*0.076383859
+ -0.5*0.000693129*T)*sT
RETURN
END
DOUBLE PRECISION FUNCTION BC( T)
implicit double precision (a-h,o-z)
DOUBLE PRECISION isT
isT=1/dsqrt(T)
BC = (((((-0.58544978*isT+0.43102052)*isT
+ +.87361369)*isT-4.13749995)*isT+2.90616279)*isT
+ -7.02181962)/T+0.02459877
RETURN
END
DOUBLE PRECISION FUNCTION BCdT( T)
implicit double precision (a-h,o-z)
DOUBLE PRECISION iST
isT=1/dsqrt(T)
BCdT = ((((-0.58544978*3.5*isT+0.43102052*3)*isT
+ +0.87361369*2.5)*isT-4.13749995*2)*isT
+ +2.90616279*1.5)*isT-7.02181962
RETURN
END
DOUBLE PRECISION FUNCTION gammaBH(X)
implicit double precision (a-h,o-z)
gammaBH=1.92907278
RETURN
END
DOUBLE PRECISION FUNCTION DALJ(T,rho)
implicit double precision (a-h,o-z)
DALJ = ((+2.01546797+rho*(-28.17881636
+ +rho*(+28.28313847+rho*(-10.42402873))))
+ +(-19.58371655+rho*(75.62340289+rho*((-120.70586598)
+ +rho*(93.92740328+rho*(-27.37737354)))))/dsqrt(T)
+ + ( (29.34470520+rho*((-112.35356937)
+ +rho*(+170.64908980+rho*((-123.06669187)
+ +rho*34.42288969))))
+ +(-13.37031968+rho*(65.38059570+
+ rho*((-115.09233113)+rho*(88.91973082
+ +rho* (-25.62099890)))))/T)/T) *rho*rho
RETURN
END

==Reference==
*[http://dx.doi.org/10.1016/0378-3812(94)80001-4 Jirí Kolafa, Ivo Nezbeda "The Lennard-Jones fluid: an accurate analytic and theoretically-based equation of state", Fluid Phase Equilibria '''100''' pp. 1-34 (1994)]
{{Source}}
{{numeric}}

Wang-Landau method

2010-11-18T13:41:30Z

Nice and Tidy: /* Extensions */ Added an internal link

The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau <ref>[http://dx.doi.org/10.1103/PhysRevLett.86.2050 Fugao Wang and D. P. Landau "Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States", Physical Review Letters '''86''' pp. 2050-2053 (2001)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevE.64.056101 Fugao Wang and D. P. Landau "Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram", Physical Review E '''64''' 056101 (2001)]</ref>
to compute the density of states, <math> \Omega (E) </math>, of [[Potts model|Potts models]];
where <math> \Omega(E) </math> is the number of [[microstate |microstates]] of the system having energy
<math> E </math>.

== Outline of the method ==
The Wang-Landau method, in its original version, is a [[Computer simulation techniques |simulation technique]] designed to achieve a uniform sampling of the energies of the system in a given range.
In a standard [[Metropolis Monte Carlo|Metropolis Monte Carlo]] in the [[canonical ensemble|canonical ensemble]]
the probability of a given [[microstate]], <math> X </math>, is given by:

:<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>;

whereas for the Wang-Landau procedure one can write:

:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;

where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes
during the simulation in order produce a predefined distribution of energies (usually
a uniform distribution); this is done by modifying the values of <math> f(E) </math>
to reduce the probability of the energies that have been already ''visited'', i.e.
If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>
is updated as:

:<math> f^{new}(E_i) = f(E_i) - \Delta f </math> ;

where it has been considered that the system has discrete values of the energy (as happens in [[Potts model|Potts Models]]), and <math> \Delta f > 0 </math>.

Such a simple scheme is continued until the shape of the energy distribution
approaches the one predefined. Notice that this simulation scheme does not produce
an equilibrium procedure, since it does not fulfill [[detailed balance]]. To overcome
this problem, the Wang-Landau procedure consists in the repetition of the scheme
sketched above along several stages. In each subsequent stage the perturbation
parameter <math> \Delta f </math> is reduced. So, for the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages can be considered as a good description of the actual equilibrium system, therefore:

:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;

where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the
[[Kronecker delta|Kronecker Delta]], and <math> g(E) </math> is the fraction of
microstates with energy <math> E </math> obtained in the sampling.

If the probability distribution of energies, <math> g(E) </math>, is nearly flat (if a uniform distribution of energies is the target), i.e.
: <math> g(E_i) \simeq 1/n_{E} ; </math>; for each value <math> E_i </math> in the selected range,
with <math> n_{E} </math> being the total number of discrete values of the energy in the range, then the density of
states will be given by:

:<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>

=== Microcanonical thermodynamics ===

Once one knows <math> \Omega(E) </math> with accuracy, one can derive the thermodynamics
of the system, since the [[entropy|entropy]] in the [[microcanonical ensemble|microcanonical ensemble]] is given by:

:<math> S \left( E \right) = k_{B} \log \Omega(E) </math>

where <math> k_{B} </math> is the [[Boltzmann constant | Boltzmann constant]].
== Extensions ==
The Wang-Landau method has inspired a number of simulation algorithms that
use the same strategy in different contexts. For example:
* [[Inverse Monte Carlo|Inverse Monte Carlo]] methods <ref>[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (2004)]</ref> <ref name="wilding">[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003)]</ref>
* [[Computation of phase equilibria]] of fluids <ref name="Lomba1">[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E '''71''' 046132 (2005)]</ref> <ref name="Lomba2">[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]</ref> <ref name="Ganzenmuller">[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]</ref>
* Control of polydispersity by [[chemical potential]] ''tuning''<ref name="wilding"> </ref>
=== Phase equilibria ===
In the original version one computes the [[entropy|entropy]] of the system as a function of
the [[internal energy|internal energy]], <math> E </math>, for fixed conditions of volume,
and number of particles.
In Refs. <ref name="Lomba1"> </ref><ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref> it was shown how the procedure can be applied to compute other thermodynamic
potentials that can be subsequently used to locate [[phase transitions]]. For instance, one
can compute the [[Helmholtz energy function | Helmholtz energy function ]],
<math> A \left( N | V, T \right) </math> as a function of the number of particle <math> N </math>
for fixed conditions of volume, <math> V </math>, and [[temperature|temperature]], <math> T </math>.
=== Refinement of the results ===
It can be convenient to supplement the Wang-Landau algorithm, which does not fulfil [[detailed balance]],
with an equilibrium simulation <ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref>. In this equilibrium simulation one can use
the final result for <math> f\left( E \right) </math> (or <math> f\left( N \right) </math>) extracted from
the Wang-Landau technique as a fixed function to weight
the probability of the different configurations.
Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,
and can be used to refine the numerical results.

==Applications==
The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.
==See also==
*[[Statistical-temperature simulation algorithm]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1119/1.1707017 D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling", American Journal of Physics '''72''' pp. 1294-1302 (2004)]
*[http://dx.doi.org/10.1063/1.2803061 R. E. Belardinelli and V. D. Pereyra "Wang-Landau algorithm: A theoretical analysis of the saturation of the error", Journal of Chemical Physics '''127''' 184105 (2007)]
*[http://dx.doi.org/10.1103/PhysRevE.75.046701 R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E '''75''' 046701 (2007)]
[[category: Monte Carlo]]
[[category: computer simulation techniques]]

Wang-Landau method

2010-11-18T10:34:05Z

Nice and Tidy:

The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau <ref>[http://dx.doi.org/10.1103/PhysRevLett.86.2050 Fugao Wang and D. P. Landau "Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States", Physical Review Letters '''86''' pp. 2050-2053 (2001)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevE.64.056101 Fugao Wang and D. P. Landau "Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram", Physical Review E '''64''' 056101 (2001)]</ref>
to compute the density of states, <math> \Omega (E) </math>, of [[Potts model|Potts models]];
where <math> \Omega(E) </math> is the number of [[microstate |microstates]] of the system having energy
<math> E </math>.

== Outline of the method ==
The Wang-Landau method, in its original version, is a [[Computer simulation techniques |simulation technique]] designed to achieve a uniform sampling of the energies of the system in a given range.
In a standard [[Metropolis Monte Carlo|Metropolis Monte Carlo]] in the [[canonical ensemble|canonical ensemble]]
the probability of a given [[microstate]], <math> X </math>, is given by:

:<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>;

whereas for the Wang-Landau procedure one can write:

:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;

where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes
during the simulation in order produce a predefined distribution of energies (usually
a uniform distribution); this is done by modifying the values of <math> f(E) </math>
to reduce the probability of the energies that have been already ''visited'', i.e.
If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>
is updated as:

:<math> f^{new}(E_i) = f(E_i) - \Delta f </math> ;

where it has been considered that the system has discrete values of the energy (as happens in [[Potts model|Potts Models]]), and <math> \Delta f > 0 </math>.

Such a simple scheme is continued until the shape of the energy distribution
approaches the one predefined. Notice that this simulation scheme does not produce
an equilibrium procedure, since it does not fulfill [[detailed balance]]. To overcome
this problem, the Wang-Landau procedure consists in the repetition of the scheme
sketched above along several stages. In each subsequent stage the perturbation
parameter <math> \Delta f </math> is reduced. So, for the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages can be considered as a good description of the actual equilibrium system, therefore:

:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;

where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the
[[Kronecker delta|Kronecker Delta]], and <math> g(E) </math> is the fraction of
microstates with energy <math> E </math> obtained in the sampling.

If the probability distribution of energies, <math> g(E) </math>, is nearly flat (if a uniform distribution of energies is the target), i.e.
: <math> g(E_i) \simeq 1/n_{E} ; </math>; for each value <math> E_i </math> in the selected range,
with <math> n_{E} </math> being the total number of discrete values of the energy in the range, then the density of
states will be given by:

:<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>

=== Microcanonical thermodynamics ===

Once one knows <math> \Omega(E) </math> with accuracy, one can derive the thermodynamics
of the system, since the [[entropy|entropy]] in the [[microcanonical ensemble|microcanonical ensemble]] is given by:

:<math> S \left( E \right) = k_{B} \log \Omega(E) </math>

where <math> k_{B} </math> is the [[Boltzmann constant | Boltzmann constant]].
== Extensions ==
The Wang-Landau method has inspired a number of simulation algorithms that
use the same strategy in different contexts. For example:
* [[Inverse Monte Carlo|Inverse Monte Carlo]] methods <ref>[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (2004)]</ref> <ref name="wilding">[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003)]</ref>
* [[Computation of phase equilibria]] of fluids <ref name="Lomba1">[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E '''71''' 046132 (2005)]</ref> <ref name="Lomba2">[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]</ref> <ref name="Ganzenmuller">[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]</ref>
* Control of polydispersity by chemical potential ''tuning''<ref name="wilding"> </ref>
=== Phase equilibria ===
In the original version one computes the [[entropy|entropy]] of the system as a function of
the [[internal energy|internal energy]], <math> E </math>, for fixed conditions of volume,
and number of particles.
In Refs. <ref name="Lomba1"> </ref><ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref> it was shown how the procedure can be applied to compute other thermodynamic
potentials that can be subsequently used to locate [[phase transitions]]. For instance, one
can compute the [[Helmholtz energy function | Helmholtz energy function ]],
<math> A \left( N | V, T \right) </math> as a function of the number of particle <math> N </math>
for fixed conditions of volume, <math> V </math>, and [[temperature|temperature]], <math> T </math>.
=== Refinement of the results ===
It can be convenient to supplement the Wang-Landau algorithm, which does not fulfil [[detailed balance]],
with an equilibrium simulation <ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref>. In this equilibrium simulation one can use
the final result for <math> f\left( E \right) </math> (or <math> f\left( N \right) </math>) extracted from
the Wang-Landau technique as a fixed function to weight
the probability of the different configurations.
Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,
and can be used to refine the numerical results.
==Applications==
The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.
==See also==
*[[Statistical-temperature simulation algorithm]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1119/1.1707017 D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling", American Journal of Physics '''72''' pp. 1294-1302 (2004)]
*[http://dx.doi.org/10.1063/1.2803061 R. E. Belardinelli and V. D. Pereyra "Wang-Landau algorithm: A theoretical analysis of the saturation of the error", Journal of Chemical Physics '''127''' 184105 (2007)]
*[http://dx.doi.org/10.1103/PhysRevE.75.046701 R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E '''75''' 046701 (2007)]
[[category: Monte Carlo]]
[[category: computer simulation techniques]]

Wang-Landau method

2010-11-18T10:31:28Z

Nice and Tidy: Changed references to Cite format

The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau <ref>[http://dx.doi.org/10.1103/PhysRevLett.86.2050 Fugao Wang and D. P. Landau "Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States", Physical Review Letters '''86''' pp. 2050-2053 (2001)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevE.64.056101 Fugao Wang and D. P. Landau "Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram", Physical Review E '''64''' 056101 (2001)]</ref>
to compute the density of states, <math> \Omega (E) </math>, of [[Potts model|Potts models]];
where <math> \Omega(E) </math> is the number of [[microstate |microstates]] of the system having energy
<math> E </math>.

== Sketches of the method ==
The Wang-Landau method, in its original version, is a [[Computer simulation techniques |simulation technique]] designed to achieve a uniform sampling of the energies of the system in a given range.
In a standard [[Metropolis Monte Carlo|Metropolis Monte Carlo]] in the [[canonical ensemble|canonical ensemble]]
the probability of a given [[microstate]], <math> X </math>, is given by:

:<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>;

whereas for the Wang-Landau procedure one can write:

:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;

where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes
during the simulation in order produce a predefined distribution of energies (usually
a uniform distribution); this is done by modifying the values of <math> f(E) </math>
to reduce the probability of the energies that have been already ''visited'', i.e.
If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>
is updated as:

:<math> f^{new}(E_i) = f(E_i) - \Delta f </math> ;

where it has been considered that the system has discrete values of the energy (as happens in [[Potts model|Potts Models]]), and <math> \Delta f > 0 </math>.

Such a simple scheme is continued until the shape of the energy distribution
approaches the one predefined. Notice that this simulation scheme does not produce
an equilibrium procedure, since it does not fulfill [[detailed balance]]. To overcome
this problem, the Wang-Landau procedure consists in the repetition of the scheme
sketched above along several stages. In each subsequent stage the perturbation
parameter <math> \Delta f </math> is reduced. So, for the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages can be considered as a good description of the actual equilibrium system, therefore:

:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;

where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the
[[Kronecker delta|Kronecker Delta]], and <math> g(E) </math> is the fraction of
microstates with energy <math> E </math> obtained in the sampling.

If the probability distribution of energies, <math> g(E) </math>, is nearly flat (if a uniform distribution of energies is the target), i.e.
: <math> g(E_i) \simeq 1/n_{E} ; </math>; for each value <math> E_i </math> in the selected range,
with <math> n_{E} </math> being the total number of discrete values of the energy in the range, then the density of
states will be given by:

:<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>

=== Microcanonical thermodynamics ===

Once one knows <math> \Omega(E) </math> with accuracy, one can derive the thermodynamics
of the system, since the [[entropy|entropy]] in the [[microcanonical ensemble|microcanonical ensemble]] is given by:

:<math> S \left( E \right) = k_{B} \log \Omega(E) </math>

where <math> k_{B} </math> is the [[Boltzmann constant | Boltzmann constant]].
== Extensions ==
The Wang-Landau method has inspired a number of simulation algorithms that
use the same strategy in different contexts. For example:
* [[Inverse Monte Carlo|Inverse Monte Carlo]] methods <ref>[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (5 pages) (2004)]</ref> <ref name="wilding">[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003)]</ref>
* [[Computation of phase equilibria]] of fluids <ref name="Lomba1">[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E '''71''' 046132 (2005)]</ref> <ref name="Lomba2">[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]</ref> <ref name="Ganzenmuller">[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]</ref>
* Control of polydispersity by chemical potential ''tuning''<ref name="wilding"> </ref>
=== Phase equilibria ===
In the original version one computes the [[entropy|entropy]] of the system as a function of
the [[internal energy|internal energy]], <math> E </math>, for fixed conditions of volume,
and number of particles.
In Refs. <ref name="Lomba1"> </ref><ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref> it was shown how the procedure can be applied to compute other thermodynamic
potentials that can be subsequently used to locate [[phase transitions]]. For instance, one
can compute the [[Helmholtz energy function | Helmholtz energy function ]],
<math> A \left( N | V, T \right) </math> as a function of the number of particle <math> N </math>
for fixed conditions of volume, <math> V </math>, and [[temperature|temperature]], <math> T </math>.
=== Refinement of the results ===
It can be convenient to supplement the Wang-Landau algorithm, which does not fulfil [[detailed balance]],
with an equilibrium simulation <ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref>. In this equilibrium simulation one can use
the final result for <math> f\left( E \right) </math> (or <math> f\left( N \right) </math>) extracted from
the Wang-Landau technique as a fixed function to weight
the probability of the different configurations.
Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,
and can be used to refine the numerical results.
==Applications==
The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.
==See also==
*[[Statistical-temperature simulation algorithm]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1119/1.1707017 D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling", American Journal of Physics '''72''' pp. 1294-1302 (2004)]
*[http://dx.doi.org/10.1063/1.2803061 R. E. Belardinelli and V. D. Pereyra "Wang-Landau algorithm: A theoretical analysis of the saturation of the error", Journal of Chemical Physics '''127''' 184105 (2007)]
*[http://dx.doi.org/10.1103/PhysRevE.75.046701 R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E '''75''' 046701 (2007)]
[[category: Monte Carlo]]
[[category: computer simulation techniques]]

Beeman's algorithm

2010-04-19T10:32:48Z

Nice and Tidy: Trivial tidy of LaTeX markup.

'''Beeman's algorithm''' <ref>[http://dx.doi.org/10.1016/0021-9991(76)90059-0 D. Beeman "Some multistep methods for use in molecular dynamics calculations", Journal of Computational Physics '''20''' pp. 130-139 (1976)]</ref> is is a method for [[Integrators for molecular dynamics |numerically integrating ordinary differential equations]], generally position and velocity, which is closely related to Verlet integration.

:<math>x(t+\Delta t) = x(t) + v(t) \Delta t + \left(\frac{2}{3}a(t) - \frac{1}{6} a(t - \Delta t) \right)\Delta t^2 + O( \Delta t^4) </math>
:<math>v(t + \Delta t) = v(t) + \left(\frac{1}{3}a(t + \Delta t) + \frac{5}{6}a(t) - \frac{1}{6}a(t - \Delta t) \right) \Delta t + O(\Delta t^3)</math>

where ''x'' is the position, ''v'' is the velocity, ''a'' is the acceleration, ''t'' is time, and <math>\Delta t</math> is the [[Time step|time-step]].

A predictor-corrector variant is useful when the forces are velocity-dependent:

:<math> x(t+\Delta t) = x(t) + v(t) \Delta t + \frac{2}{3}a(t) \Delta t^2 - \frac{1}{6} a(t - \Delta t) \Delta t^2 + O( \Delta t^4).</math>

The velocities at time <math>t =t + \Delta t</math> are then calculated from the positions.

:<math> v(t + \Delta t)_{(\mathrm{predicted})} = v(t) + \frac{3}{2}a(t) \Delta t - \frac{1}{2}a(t - \Delta t) \Delta t + O( \Delta t^3)</math>

The accelerations at time <math>t =t + \Delta t</math> are then calculated from the positions and predicted velocities.

:<math> v(t + \Delta t)_{(\mathrm{corrected})} = v(t) + \frac{1}{3}a(t + \Delta t) \Delta t + \frac{5}{6}a(t) \Delta t - \frac{1}{6}a(t - \Delta t) \Delta t + O( \Delta t^3) </math>

==See also==
*[[Velocity Verlet algorithm]]

==References==
<references/>
==External links==
*[http://en.wikipedia.org/wiki/Beeman%27s_algorithm Beeman's algorithm entry on wikipedia]
[[category: Molecular dynamics]]

Mathematics

2010-03-05T13:40:08Z

Nice and Tidy: /* General */ Added an internal link.

:'' "Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach, Hertz, Boltzmann and Volkmann. It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also." ''
::::: '''David Hilbert''' (excerpt from problem #6 of his lecture ''"Mathematical Problems"'', delivered before the Second International Congress of Mathematicians in Paris, 1900)
Here are details of mathematical tools that have applications in [[classical thermodynamics]], [[statistical mechanics]] and [[computer simulation techniques]].
==General==
*[[Clebsch-Gordan coefficients]]
*[[Euler angles]]
*[[Gauss theorem]]
*[[Green's theorem]]
*[[Lagrange multipliers]]
*[[Laplace's equation]]
*[[Lee-Yang circle theorem]]
*[[Linear algebra]]
*[[Liouville's theorem]]
*[[Lyapunov exponents]]
*[[Padé approximants]]
*[[Perron-Frobenius theorem]]
*[[Poincaré theorem]]
*[[Power series]]
*[[Predictor-corrector methods]]
*[[Random numbers]]
*[[Root finding]]
*[[Runge-Kutta method]]
*[[Spherical harmonics]]
*[[Stirling's approximation]]
*[[Taylor expansion]]
*[[Trotter formula]]
*[[Wigner D-matrix]]
*[[Wigner D-matrix |Wigner rotation matrices]]

==Functions and distributions==
*[[Bessel functions]]
*[[Dirac delta distribution]]
*[[Gamma function]]
*[[Gaussian distribution]]
*[[Heaviside step distribution]]
*[[Kronecker delta]]
*[[Poisson distribution]]
*[[Ramp function]]

==Geometry==
*[[Alpha shape]]
*[[Delaunay simplexes]]
*[[Dual lattice]]
*[[Lattice Structures]]
*[[Voronoi cells]]
==Integrals and quadrature==
*[[Standard integrals]]
*[[Elliptic integrals]]
*[[Newton-Cotes formulas]]
*[[Gauss-Chebyshev quadrature]]
*[[Gauss-Legendre quadrature]]
==Polynomials==
*[[Chebyshev polynomials]]
*[[Laguerre polynomials]]
*[[Legendre polynomials]]
**[[Associated Legendre functions]]

==Transforms==
*[[Fast Fourier transform]]
*[[Fourier analysis]]
*[[Laplace transform]]
*[[Legendre transform]]
==Computer assisted mathematics==
{| class="wikitable sortable" border="1" cellpadding="5" cellspacing="0"
|-
! Packages !! Program or library !! class="unsortable" | Notes !! License
|-
|[[BLAS]]
|routines
| '''B'''asic '''L'''inear '''A'''lgebra '''S'''ubprograms
|free source
|-
|[[CGAL]]
|library
|'''C'''omputational '''G'''eometry '''A'''lgorithms '''L'''ibrary
|free, Open Source
|-
|[[IMSL Numerical Libraries]]
|library
|
|
|-
|[[LAPACK]]
|routines
|'''L'''inear '''A'''lgebra '''PACK'''age
|free source
|-
|[[ScaLAPACK]]
|library
|'''Sca'''lable '''LAPACK'''
|free source
|-
|[[Magma]]
|
|
|
|-
|[[Maple]]
|program
|
|commercial
|-
|[[Mathematica]]
|program
|
|commercial
|-
|[[Math Kernel Library]]
|library
|
|
|-
|[[MATLAB]]
|program
|
|commercial
|-
|[[Maxima]]
|program
|
|free (General Public License)
|-
|[[NAG]]
|routines
|'''N'''umerical '''A'''lgorithms '''G'''roup
|commercial
|-
|[[Numerical Recipes]]
|routines
|
|
|-
|[[Octave]]
|program
|
|free (GNU)
|-
|[[QUADPACK]]
|routines
|
|free
|-
|[[SAGE]]
|program
|
|free (General Public License)
|-
|[[Scilab]]
|program
|
|free, Open Source
|}

==Interesting reading==
*[http://dx.doi.org/10.1002/cpa.3160130102 Eugene P. Wigner "The unreasonable effectiveness of mathematics in the natural sciences", Communications on Pure and Applied Mathematics '''13''' pp. 1-14 (1960)]
**[http://dx.doi.org/10.1063/1.2435629 Frank Wilczek "Reasonably Effective: I. Deconstructing a Miracle", Physics Today '''November''' pp. 8-9 (2006)]
**[http://dx.doi.org/10.1063/1.2743102 Frank Wilczek "Reasonably Effective: II. Devil's Advocate", Physics Today '''May''' pp. 8-9 (2007)]
*[http://www.ams.org/bull/1988-19-01/S0273-0979-1988-15634-0/S0273-0979-1988-15634-0.pdf David Ruelle "Is our mathematics natural? The case of equilibrium statistical mechanics", Bulletin of the American Mathematical Society '''19''' pp. 259-268 (1988)]
[[category: mathematics]]

Inverse temperature

2010-03-05T13:39:23Z

Nice and Tidy: Slight tidy.

It is often convenient to define a dimensionless '''inverse temperature''', <math>\beta</math>:

:<math>\beta := \frac{1}{k_BT}</math>

This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.
Indeed, it shown in Ref. 1 (pp. 79-85) that this is the way it enters. The task is to maximize number of ways <math>N</math> particles may be assigned to <math>K</math> space-momentum cells, such that one has a set of occupation numbers <math>n_i</math>. Introducing the [[partition function]]:

:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>

one could maximize its logarithm (a monotonous function):

:<math>\log \Omega \approx \log N -N - \sum_ i ( \log n_i + n_i) + \mathrm{consts} ,</math>

where [[Stirling's approximation]] for large numbers has been used. The maximization must be performed subject to the constraint:

:<math>\sum_i n_i=N</math>

An additional constraint, which applies only to dilute gases, is:

:<math>\sum_i n_i e_i=E, </math>

where <math>E</math> is the total energy and <math>e_i=p_i^2/2m</math> is the energy of cell <math>i</math>.
The method of [[Lagrange multipliers]] entails finding the extremum of the function

:<math>L=\log\Omega - \alpha (\sum_i n_i - N ) - \beta ( \sum_i n_i e_i - E ),</math>

where the two Lagrange multipliers enforce the two conditions and permit the treatment of
the occupations as independent variables. The minimization leads to

:<math>n_i=C e^{-\beta e_i}, </math>

and an application to the case of an ideal gas reveals the connection with the temperature,

:<math>\beta := \frac{1}{k_BT} .</math>

Similar methods are used for [[quantum statistics]] of dilute gases (Ref. 1, pp. 179-185).
==References==
#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1
[[category: Classical thermodynamics]]
[[category: statistical mechanics]]
[[category: Non-equilibrium thermodynamics]]

VLABON force field

2010-03-05T13:20:06Z

Nice and Tidy: New page: {{Stub-general}} '''VLABON''' force field <ref>[http://dx.doi.org/10.1021/ja00063a043 Daniel M. Root, Clark R. Landis, Thomas Cleveland "Valence bond concepts applied to...

{{Stub-general}}
'''VLABON''' [[force fields | force field]] <ref>[http://dx.doi.org/10.1021/ja00063a043 Daniel M. Root, Clark R. Landis, Thomas Cleveland "Valence bond concepts applied to the molecular mechanics description of molecular shapes. 1. Application to nonhypervalent molecules of the P-block", Journal of the American Chemical Society '''115''' pp. 4201-4209 (1993)]</ref> <ref>[http://dx.doi.org/10.1021/ja9506521 Thomas Cleveland and Clark R. Landis "Valence Bond Concepts Applied to the Molecular Mechanics Description of Molecular Shapes. 2. Applications to Hypervalent Molecules of the P-Block", Journal of the American Chemical Society '''118''' pp. 6020-6030 (1996)]</ref>.
==Functional form==
==Parameters==
==References==
<references/>
[[category:force fields]]

SHAPES force field

2010-03-05T13:14:20Z

Nice and Tidy: New page: {{Stub-general}} '''SHAPES''' force field <ref>[http://dx.doi.org/10.1021/ja00001a001 Viloya S. Allured, Christine M. Kelly, Clark R. Landis "SHAPES empirical force fiel...

{{Stub-general}}
'''SHAPES''' [[force fields | force field]] <ref>[http://dx.doi.org/10.1021/ja00001a001 Viloya S. Allured, Christine M. Kelly, Clark R. Landis "SHAPES empirical force field: new treatment of angular potentials and its application to square-planar transition-metal complexes", Journal of the American Chemical Society '''113''' pp. 1-12 (1991)]</ref>.
==Functional form==
==Parameters==
==References==
<references/>
[[category:force fields]]

Force fields

2010-03-05T13:12:08Z

Nice and Tidy: /* List of force fields */

'''Force fields''' consist of (hopefully) transferable parameters for molecular sub-units. They are designed to be applicable to a variety of molecular systems, in particular for [[flexible molecules]] , over a range of thermodynamic conditions.
==List of force fields==
{{columns-list|3|
*[[Force fields: Alkanes in nanoporous materials | Alkanes in nanoporous materials]]
*[[AMBER forcefield |AMBER]]
*[[AMBERN]]
*[[AMOEBA]]
*[[Approximate pair theory]]
*[[CFF]]
*[[CFF91]]
*[[CHARMM]]
*[[CLAYFF]]
*[[COMPASS]]
*[[CVFF]]
*[[DREIDING]]
*[[ECEPP/2]]
*[[ECEPP/3]]
*[[ENCAD]]
*[[GAFF]]
*[[GROMOS]]
*[[HFF]]
*[[LCFF]]
*[[MARTINI]]
*[[MM2]]
*[[MM3]]
*[[MM4]]
*[[MMFF94]]
*[[MVFF]]
*[[NERD]]
*[[OPLS]]
*[[PCFF]]
*[[poly(ethylene oxide)]]
*[[ReaxFF]]
*[[SHAPES force field |SHAPES]]
*[[SYBYL]]
*[[TraPPE]]
*[[TRIPOS]]
*[[UFF]]
*[[UNRES]]
*[[VLABON force field | VALBON]]
*[[VFF]]
*[[WBFF]]
}}

==See also==
*[[Idealised models]]
*[[Realistic models]] as well as [[polymers]] and [[proteins]].
[[category: Computer simulation techniques]]

Force fields

2010-03-05T13:07:05Z

Nice and Tidy: /* List of force fields */

'''Force fields''' consist of (hopefully) transferable parameters for molecular sub-units. They are designed to be applicable to a variety of molecular systems, in particular for [[flexible molecules]] , over a range of thermodynamic conditions.
==List of force fields==
{{columns-list|3|
*[[Force fields: Alkanes in nanoporous materials | Alkanes in nanoporous materials]]
*[[AMBER forcefield |AMBER]]
*[[AMBERN]]
*[[AMOEBA]]
*[[Approximate pair theory]]
*[[CFF]]
*[[CFF91]]
*[[CHARMM]]
*[[CLAYFF]]
*[[COMPASS]]
*[[CVFF]]
*[[DREIDING]]
*[[ECEPP/2]]
*[[ECEPP/3]]
*[[ENCAD]]
*[[GAFF]]
*[[GROMOS]]
*[[HFF]]
*[[LCFF]]
*[[MARTINI]]
*[[MM2]]
*[[MM3]]
*[[MM4]]
*[[MMFF94]]
*[[MVFF]]
*[[NERD]]
*[[OPLS]]
*[[PCFF]]
*[[poly(ethylene oxide)]]
*[[ReaxFF]]
*[[SHARP force field |SHARP]]
*[[SYBYL]]
*[[TraPPE]]
*[[TRIPOS]]
*[[UFF]]
*[[UNRES]]
*[[VLABON force field | VALBON]]
*[[VFF]]
*[[WBFF]]
}}

==See also==
*[[Idealised models]]
*[[Realistic models]] as well as [[polymers]] and [[proteins]].
[[category: Computer simulation techniques]]

WBFF force field

2010-03-05T13:04:25Z

Nice and Tidy:

{{Stub-general}}
'''WBFF''' (White-Bovill [[Force fields | Force-Field]]) <ref>[http://dx.doi.org/10.1039/P29770001610 David N. J. White and Moira J. Bovill "Molecular mechanics calculations on alkanes and non-conjugated alkenes", Journal of the Chemical Society, Perkin Transactions 2 pp. 1610-1623 (1977)]</ref> is designed for molecular mechanics calculations of [[Realistic models#Alkanes |alkanes]], specifically for H, C(''sp''2) and C(''sp''3).
==Functional form==
==Parameters==
==References==
<references/>
[[Category: Force fields]]

WBFF

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Nice and Tidy: WBFF moved to WBFF force field

#REDIRECT [[WBFF force field]]

WBFF force field

2010-03-05T13:04:16Z

Nice and Tidy: WBFF moved to WBFF force field

{{Stub-general}}
'''WBFF''' (White-Bovill [[Force fields | Force-Field]]) <ref>[http://dx.doi.org/10.1039/P29770001610 David N. J. White and Moira J. Bovill "Molecular mechanics calculations on alkanes and non-conjugated alkenes", Journal of the Chemical Society, Perkin Transactions 2 pp. 1610-1623 (1977)]</ref> is designed for molecular mechanics calculations of [[Realistic models#Alkanes |alkanes]], specifically for H, C(''sp''2) and C(''sp''3).

==References==
<references/>
[[Category: Force fields]]

UNRES force field

2010-03-05T13:04:02Z

Nice and Tidy:

{{Stub-general}}
'''UNRES''' is a united-residue [[force fields |force field]] for energy-based prediction of [[proteins |protein]] structure.
==Functional form==
==Parameters==
==References==
#[http://www.proteinscience.org/cgi/content/abstract/2/10/1697 A. LIWO, M. R. PINCUS, R. J. WAWAK, S. RACKOVSKY and H. A. SCHERAGA "Calculation of protein backbone geometry from {alpha}-carbon coordinates based on peptide-group dipole alignment", Protein Science '''2''' pp. 1697-1714 (1993)]
#[http://www.pnas.org/cgi/content/abstract/96/10/5482 Adam Liwo, Jooyoung Lee, Daniel R. Ripoll, Jaroslaw Pillardy, and Harold A. Scheraga "Protein structure prediction by global optimization of a potential energy function", PNAS '''96''' pp.5482-5485 (1999)]
==External links==
*[http://www.chem.univ.gda.pl/~adam/local/docs/index.htm Parameters of the UNRES force field]
[[Category: Force fields]]

UNRES

2010-03-05T13:03:51Z

Nice and Tidy: UNRES moved to UNRES force field

#REDIRECT [[UNRES force field]]

UNRES force field

2010-03-05T13:03:51Z

Nice and Tidy: UNRES moved to UNRES force field

{{Stub-general}}
'''UNRES''' is a united-residue [[force fields |force field]] for energy-based prediction of [[proteins |protein]] structure.
==References==
#[http://www.proteinscience.org/cgi/content/abstract/2/10/1697 A. LIWO, M. R. PINCUS, R. J. WAWAK, S. RACKOVSKY and H. A. SCHERAGA "Calculation of protein backbone geometry from {alpha}-carbon coordinates based on peptide-group dipole alignment", Protein Science '''2''' pp. 1697-1714 (1993)]
#[http://www.pnas.org/cgi/content/abstract/96/10/5482 Adam Liwo, Jooyoung Lee, Daniel R. Ripoll, Jaroslaw Pillardy, and Harold A. Scheraga "Protein structure prediction by global optimization of a potential energy function", PNAS '''96''' pp.5482-5485 (1999)]
==External links==
*[http://www.chem.univ.gda.pl/~adam/local/docs/index.htm Parameters of the UNRES force field]
[[Category: Force fields]]

UFF force field

2010-03-05T13:03:34Z

Nice and Tidy:

{{Stub-general}}
'''Universal Force Field''' (UFF)
==Functional form==
==Parameters==
==References==
#[http://dx.doi.org/10.1021/ja00051a040 A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard, and W. M. Skiff "UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations", Journal of the American Chemical Society '''114''' pp. 10024 - 10035 (1992)]
#[http://dx.doi.org/10.1021/j100161a070 Anthony K. Rappe and William A. Goddard "Charge equilibration for molecular dynamics simulations",Journal of Physical Chemistry '''95''' pp. 3358 - 3363 (1991)]
[[category: force fields]]

UFF

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#REDIRECT [[UFF force field]]

UFF force field

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{{Stub-general}}
'''Universal Force Field''' (UFF)
==References==
#[http://dx.doi.org/10.1021/ja00051a040 A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard, and W. M. Skiff "UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations", Journal of the American Chemical Society '''114''' pp. 10024 - 10035 (1992)]
#[http://dx.doi.org/10.1021/j100161a070 Anthony K. Rappe and William A. Goddard "Charge equilibration for molecular dynamics simulations",Journal of Physical Chemistry '''95''' pp. 3358 - 3363 (1991)]
[[category: force fields]]

TraPPE

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#REDIRECT [[TraPPE force field]]

TraPPE force field

2010-03-05T13:02:39Z

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The set of '''Transferable Potentials for Phase Equilibria''' (TraPPE) [[Force fields |force-field]] was developed for the simulation of linear and branched [[Realistic models#Alkanes |alkanes]], [[Realistic models#Alcohols |alcohols]], ethers, ketones, and aldehydes.
It is based upon the previous [[OPLS]] force field.
==TraPPE-UA==
This is the [[United-atom model |united atom]] version of TraPPE
==TraPPE–EH==
TraPPE publications 3 and 9 provide data for explicit hydrogen simulations.
==Parameters==
The following 9 publications provide the TraPPE parameters:
#[http://dx.doi.org/10.1021/jp972543+ Marcus G. Martin and J. Ilja Siepmann "Transferable potentials for phase equilibria. 1. United-atom description of n-alkanes,", The Journal of Physical Chemistry B '''102''', pp. 2569-2577 (1998)]
#[http://dx.doi.org/10.1021/jp984742e Marcus G. Martin and J. Ilja Siepmann "Novel Configurational-Bias Monte Carlo Method for Branched Molecules. Transferable Potentials for Phase Equilibria. 2. United-Atom Description of Branched Alkanes", The Journal of Physical Chemistry B '''103''' pp. 4508 -4517 (1999)]
#[http://dx.doi.org/10.1021/jp990822m Bin Chen and J. Ilja Siepmann "Transferable Potentials for Phase Equilibria. 3. Explicit-Hydrogen Description of Normal Alkanes", The Journal of Physical Chemistry B '''103''' pp. 5370 -5379 (1999)]
#[http://dx.doi.org/10.1021/jp001044x Collin D. Wick, Marcus G. Martin, and J. Ilja Siepmann "Transferable Potentials for Phase Equilibria. 4. United-Atom Description of Linear and Branched Alkenes and Alkylbenzenes", The Journal of Physical Chemistry B '''104''' pp. 8008 -8016 (2000)]
#[http://dx.doi.org/10.1021/jp003882x Bin Chen, Jeffrey J. Potoff, and J. Ilja Siepmann "Monte Carlo Calculations for Alcohols and Their Mixtures with Alkanes. Transferable Potentials for Phase Equilibria. 5. United-Atom Description of Primary, Secondary, and Tertiary Alcohols", The Journal of Physical Chemistry B '''105''' pp. 3093 -3104 (2001)]
#[http://dx.doi.org/10.1021/jp049459w John M. Stubbs, Jeffrey J. Potoff, and J. Ilja Siepmann "Transferable Potentials for Phase Equilibria. 6. United-Atom Description for Ethers, Glycols, Ketones, and Aldehydes", The Journal of Physical Chemistry B '''106''' pp. 17596 -17605 (2004)]
#[http://dx.doi.org/10.1021/jp0504827 Collin D. Wick, John M. Stubbs, Neeraj Rai, and J. Ilja Siepmann "Transferable Potentials for Phase Equilibria. 7. Primary, Secondary, and Tertiary Amines, Nitroalkanes and Nitrobenzene, Nitriles, Amides, Pyridine, and Pyrimidine", The Journal of Physical Chemistry B '''109''' pp. 18974 -18982 (2005)]
#[http://dx.doi.org/10.1021/jp0549125 Nusrat Lubna, Ganesh Kamath, Jeffrey J. Potoff, Neeraj Rai, and J. Ilja Siepmann "Transferable Potentials for Phase Equilibria. 8. United-Atom Description for Thiols, Sulfides, Disulfides, and Thiophene", The Journal of Physical Chemistry B '''109''' pp. 24100 -24107 (2005)]
#[http://dx.doi.org/10.1021/jp073586l Neeraj Rai and J. Ilja Siepmann "Transferable Potentials for Phase Equilibria. 9. Explicit Hydrogen Description of Benzene and Five-Membered and Six-Membered Heterocyclic Aromatic Compounds", The Journal of Physical Chemistry B '''111''' pp. 10790 -10799 (2007)]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1080/0892702031000117270 Josep C. Pamies, Clare McCabe, Peter T. Cummings and Lourdes F. Vega "Coexistence Densities of Methane and Propane by Canonical Molecular Dynamics and Gibbs Ensemble Monte Carlo Simulations", Molecular Simulation '''29''' pp. 463-470 (2003)]
==External links==
*[http://www.chem.umn.edu/groups/siepmann/trappe/intro.php TraPPE's pages at Siepmann's group at the University of Minnesota]
[[category:force fields]]

ReaxFF force field

2010-03-05T13:02:07Z

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{{Stub-general}}
The '''ReaxFF''' force-field was developed by the [http://www.wag.caltech.edu/home/wag/index.html Prof. William A. Goddard, III] group: [http://www.wag.caltech.edu/ Materials and Process Simulation Center] located at the California Institute of Technology.
==Functional form==
==Parameters==
==References==
#[http://dx.doi.org/10.1021/jp004368u Adri C. T. van Duin, Siddharth Dasgupta, Francois Lorant, and William A. Goddard, III "ReaxFF: A Reactive Force Field for Hydrocarbons", The Journal of Physical Chemistry A (Molecules) '''105''' pp. 9396 - 9409 (2001)]
[[category: force fields]]

ReaxFF

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#REDIRECT [[ReaxFF force field]]

ReaxFF force field

2010-03-05T13:01:57Z

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{{Stub-general}}
The '''ReaxFF''' force-field was developed by the [http://www.wag.caltech.edu/home/wag/index.html Prof. William A. Goddard, III] group: [http://www.wag.caltech.edu/ Materials and Process Simulation Center] located at the California Institute of Technology.
==References==
#[http://dx.doi.org/10.1021/jp004368u Adri C. T. van Duin, Siddharth Dasgupta, Francois Lorant, and William A. Goddard, III "ReaxFF: A Reactive Force Field for Hydrocarbons", The Journal of Physical Chemistry A (Molecules) '''105''' pp. 9396 - 9409 (2001)]
[[category: force fields]]

PCFF force field

2010-03-05T13:01:37Z

Nice and Tidy:

'''PCFF''' is a member of the consistent family of force fields ([[CFF91]], PCFF, [[CFF]] and [[COMPASS]]), which are closely related second-generation [[force fields]]. They were parameterized against a wide range of experimental observables for organic compounds containing [[hydrogen |H]], [[carbon |C]], [[nitrogen |N]], [[oxygen |O]], [[Sulfur |S]], [[Phosphorus |P]], halogen atoms and ions, alkali metal cations, and several biochemically important divalent metal cations. PCFF is based on CFF91, extended so as to have a broad coverage of organic polymers, (inorganic) metals, and [[zeolites]]. It was developed based on CFF91 and its is intended for application to [[polymers]] and organic materials. It is useful for polycarbonates, melamine resins, polysaccharides, other polymers, organic and inorganic materials, about 20 inorganic metals, as well as for carbohydrates, [[lipids]], and [[nucleic acids]] in calculations of cohesive energies, mechanical properties, [[Compressibility |compressibilities]], [[Heat capacity |heat capacities]], [[elastic constants]]. It handles electron delocalization in aromatic rings by means of a charge library rather than bond increments.
==Functional form==
==Parameters==
==References==
#[http://dx.doi.org/10.1021/ja00086a030 Huai Sun, Stephen J. Mumby, Jon R. Maple, Arnold T. Hagler "An ab Initio CFF93 All-Atom Force Field for Polycarbonates", Journal of the American Chemical Society '''116''' pp. 2978–2987 (1994)]
[[category: force fields]]

PCFF

2010-03-05T13:01:23Z

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#REDIRECT [[PCFF force field]]

PCFF force field

2010-03-05T13:01:23Z

Nice and Tidy: PCFF moved to PCFF force field

'''PCFF''' is a member of the consistent family of force fields ([[CFF91]], PCFF, [[CFF]] and [[COMPASS]]), which are closely related second-generation [[force fields]]. They were parameterized against a wide range of experimental observables for organic compounds containing [[hydrogen |H]], [[carbon |C]], [[nitrogen |N]], [[oxygen |O]], [[Sulfur |S]], [[Phosphorus |P]], halogen atoms and ions, alkali metal cations, and several biochemically important divalent metal cations. PCFF is based on CFF91, extended so as to have a broad coverage of organic polymers, (inorganic) metals, and [[zeolites]]. It was developed based on CFF91 and its is intended for application to [[polymers]] and organic materials. It is useful for polycarbonates, melamine resins, polysaccharides, other polymers, organic and inorganic materials, about 20 inorganic metals, as well as for carbohydrates, [[lipids]], and [[nucleic acids]] in calculations of cohesive energies, mechanical properties, [[Compressibility |compressibilities]], [[Heat capacity |heat capacities]], [[elastic constants]]. It handles electron delocalization in aromatic rings by means of a charge library rather than bond increments.
==References==
#[http://dx.doi.org/10.1021/ja00086a030 Huai Sun, Stephen J. Mumby, Jon R. Maple, Arnold T. Hagler "An ab Initio CFF93 All-Atom Force Field for Polycarbonates", Journal of the American Chemical Society '''116''' pp. 2978–2987 (1994)]
[[category: force fields]]

OPLS

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#REDIRECT [[OPLS force field]]

OPLS force field

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{{Stub-general}}
The '''optimized potentials for liquid simulations''' (OPLS) [[Force fields |force-field]] was developed for the simulation
of [[proteins]] and other organic liquids.
====References====
#[http://dx.doi.org/10.1021/ja00214a001 William L. Jorgensen and Julian Tirado-Rives "The OPLS (optimized potentials for liquid simulations) potential functions for proteins, energy minimizations for crystals of cyclic peptides and crambin", Journal of the American Chemical Society '''110''' pp. 1657 - 1666 (1988)]
#[http://dx.doi.org/10.1002/jcc.540140207 William L. Jorgensen and Toan B. Nguyen "Monte Carlo simulations of the hydration of substituted benzenes with OPLS potential functions", Journal of Computational Chemistry '''14''' pp. 195 - 205 (1993)]
==OPLS-all atom==
(OPLS-AA)
====Form of the force field====
Bond stretching
:<math>E_{\rm {bond}} = \sum_{\rm {bonds}} K_r \left(r-r_{eq}\right)^2</math>
Angle bending
:<math>E_{\rm {angle}} = \sum_{\rm {angles}} K_\theta \left(\theta-\theta_{eq}\right)^2</math>
Torsion
:<math>E(\phi) = \frac{V_1}{2} \left[ 1 + \cos (\phi +f1)\right] + \frac{V_2}{2} \left[ 1 - \cos (2\phi +f2)\right] + \frac{V_3}{2} \left[ 1 + \cos (3\phi +f3)\right]</math>
Non-bonded
:<math>E_{ab} = \sum_i^{\rm {on~a}}\sum_j^{\rm {on~b}} \left[ \frac{q_i q_j e^2}{r_{ij}} + 4\epsilon_{ij} \left( \frac{\sigma_{ij}^{12}}{r_{ij}^{12}}-\frac{\sigma_{ij}^{6}}{r_{ij}^{6}}\right)\right]f_{ij}, ~~~~f_{ij} = \left\{ \begin{array}{lll}
0.5 & ; & {\rm {if}}~~ i,j = 1,4 \\
1 & ; & {\rm {otherwise}} \end{array} \right. </math>
====References====
#[http://www3.interscience.wiley.com/cgi-bin/abstract/48444/ABSTRACT Wolfgang Damm, Antonio Frontera, Julian Tirado-Rives and William L. Jorgensen "OPLS all-atom force field for carbohydrates", Journal of Computational Chemistry '''18''' pp. 1955 - 1970 (1997)]
#[http://dx.doi.org/10.1021/ja9621760 William L. Jorgensen, David S. Maxwell, and Julian Tirado-Rives "Development and Testing of the OPLS All-Atom Force Field on Conformational Energetics and Properties of Organic Liquids", Journal of the American Chemical Society '''118''' pp. 11225 - 11236 (1996)]
##[http://pubs.acs.org/subscribe/journals/jacsat/suppinfo/118/i45/ja9621760/ja11225.pdf Electronic Supporting Information]
[[category:force fields]]

NERD force field

2010-03-05T13:00:12Z

Nice and Tidy:

The '''NERD''' force field for [[Realistic models#Alkanes |alkanes]] was developed by '''N'''ath, '''E'''scobedo, and '''d'''e Pablo (Ref. 1) (with the addition of the word '''R'''evised one arrives at NERD, see reference 15 within Ref. 1. The amusing nature of the name in English was far from lost on the authors). The NERD
parameters show a certain similitude to other alkane parameter sets, such as those of López Rodríguez and co-workers (Refs. 2-4) or those of Smit, Karaborni, and Siepmann (SKS) (Refs. 5-7).
==Parameters==
In the NERD model the non-bonded [[intermolecular pair potential]] is that of the [[Lennard-Jones model]] with the following parameters (Ref. 1 Table I):
:{| border="1"
|-
| Molecule || <math>\sigma_{\mathrm {CH}_3}</math> || <math>\sigma_{\mathrm {CH}_2}</math> || <math>\epsilon_{\mathrm {CH}_3}</math> || <math>\epsilon_{\mathrm {CH}_2}</math>
|-
| [[ethane]] || 3.825 <math>\mathrm{\AA}</math> || 3.93 <math>\mathrm{\AA}</math> || 100.6 K
|-
| [[propane]] || 3.857 <math>\mathrm{\AA}</math> || 3.93 <math>\mathrm{\AA}</math> || 102.6 K || 45.8 K
|-
| [[butane]] and longer || 3.91 <math>\mathrm{\AA}</math> || 3.93 <math>\mathrm{\AA}</math> || 104.0 K
|| 45.8 K
|}

==References==
#[http://dx.doi.org/10.1063/1.476429 Shyamal K. Nath, Fernando A. Escobedo, and Juan J. de Pablo "On the simulation of vapor–liquid equilibria for alkanes", Journal of Chemical Physics '''108''' pp. 9905- (1998)]
#[http://dx.doi.org/10.1080/00268979100101471 Antonio López Rodríguez, Carlos Vega, Juan J. Freire and Santiago Lago "Potential parameters of methyl and methylene obtained from second virial coefficients of n-alkanes", Molecular Physics '''73''' pp. 691-701 (1991)]
#[http://dx.doi.org/10.1063/1.472291 Carlos Vega and Antonio López Rodríguez "Second virial coefficients, critical temperatures, and the molecular shapes of long n-alkanes", Journal of Chemical Physics '''105''' pp. 4223- (1996)]
#[http://dx.doi.org/10.1063/1.479283 A. Lopez Rodriguez, C. Vega and J. J. Freire "Determination of potential parameters for alkanes", Journal of Chemical Physics '''111''' pp. 438- (1999)]
#[http://dx.doi.org/10.1063/1.469563 Berend Smit, Sami Karaborni, and J. Ilja Siepmann "Computer simulations of vapor–liquid phase equilibria of n-alkanes", Journal of Chemical Physics '''102''' pp. 2126- (1995)]
##[http://dx.doi.org/10.1063/1.476536 Berend Smit, Sami Karaborni, and J. Ilja Siepmann "Erratum: "Computer simulations of vapor–liquid phase equilibria of n-alkanes"", Journal of Chemical Physics '''109''' p. 352 (1998)]
#[http://dx.doi.org/10.1021/ja00067a088 J. Ilja Siepmann, Sami Karaborni, and Berend Smit "Vapor-liquid equilibria of model alkanes", Journal of the American Chemical Society '''115''' pp. 6454-6455 (1993)]
#[http://dx.doi.org/10.1038/365330a0 J. IIja Siepmann, Sami Karaborni and Berend Smit "Simulating the critical behaviour of complex fluids", Nature '''365''' pp. 330-332 (1993)]
[[category:Force fields]]

NERD

2010-03-05T12:59:38Z

Nice and Tidy: NERD moved to NERD force field

#REDIRECT [[NERD force field]]

NERD force field

2010-03-05T12:59:38Z

Nice and Tidy: NERD moved to NERD force field

The '''NERD''' force field for [[Realistic models#Alkanes |alkanes]] was developed by '''N'''ath, '''E'''scobedo, and '''d'''e Pablo (Ref. 1) (with the addition of the word '''R'''evised one arrives at NERD, see reference 15 within Ref. 1. The amusing nature of the name in English was far from lost on the authors). The NERD
parameters show a certain similitude to other alkane parameter sets, such as those of López Rodríguez and co-workers (Refs. 2-4) or those of Smit, Karaborni, and Siepmann (SKS) (Refs. 5-7),
In the NERD model the non-bonded [[intermolecular pair potential]] is that of the [[Lennard-Jones model]] with the following parameters (Ref. 1 Table I):
:{| border="1"
|-
| Molecule || <math>\sigma_{\mathrm {CH}_3}</math> || <math>\sigma_{\mathrm {CH}_2}</math> || <math>\epsilon_{\mathrm {CH}_3}</math> || <math>\epsilon_{\mathrm {CH}_2}</math>
|-
| [[ethane]] || 3.825 <math>\mathrm{\AA}</math> || 3.93 <math>\mathrm{\AA}</math> || 100.6 K
|-
| [[propane]] || 3.857 <math>\mathrm{\AA}</math> || 3.93 <math>\mathrm{\AA}</math> || 102.6 K || 45.8 K
|-
| [[butane]] and longer || 3.91 <math>\mathrm{\AA}</math> || 3.93 <math>\mathrm{\AA}</math> || 104.0 K
|| 45.8 K
|}

==References==
#[http://dx.doi.org/10.1063/1.476429 Shyamal K. Nath, Fernando A. Escobedo, and Juan J. de Pablo "On the simulation of vapor–liquid equilibria for alkanes", Journal of Chemical Physics '''108''' pp. 9905- (1998)]
#[http://dx.doi.org/10.1080/00268979100101471 Antonio López Rodríguez, Carlos Vega, Juan J. Freire and Santiago Lago "Potential parameters of methyl and methylene obtained from second virial coefficients of n-alkanes", Molecular Physics '''73''' pp. 691-701 (1991)]
#[http://dx.doi.org/10.1063/1.472291 Carlos Vega and Antonio López Rodríguez "Second virial coefficients, critical temperatures, and the molecular shapes of long n-alkanes", Journal of Chemical Physics '''105''' pp. 4223- (1996)]
#[http://dx.doi.org/10.1063/1.479283 A. Lopez Rodriguez, C. Vega and J. J. Freire "Determination of potential parameters for alkanes", Journal of Chemical Physics '''111''' pp. 438- (1999)]
#[http://dx.doi.org/10.1063/1.469563 Berend Smit, Sami Karaborni, and J. Ilja Siepmann "Computer simulations of vapor–liquid phase equilibria of n-alkanes", Journal of Chemical Physics '''102''' pp. 2126- (1995)]
##[http://dx.doi.org/10.1063/1.476536 Berend Smit, Sami Karaborni, and J. Ilja Siepmann "Erratum: "Computer simulations of vapor–liquid phase equilibria of n-alkanes"", Journal of Chemical Physics '''109''' p. 352 (1998)]
#[http://dx.doi.org/10.1021/ja00067a088 J. Ilja Siepmann, Sami Karaborni, and Berend Smit "Vapor-liquid equilibria of model alkanes", Journal of the American Chemical Society '''115''' pp. 6454-6455 (1993)]
#[http://dx.doi.org/10.1038/365330a0 J. IIja Siepmann, Sami Karaborni and Berend Smit "Simulating the critical behaviour of complex fluids", Nature '''365''' pp. 330-332 (1993)]
[[category:Force fields]]

MMFF94 force field

2010-03-05T12:59:17Z

Nice and Tidy:

{{Stub-general}}
The '''Merck molecular force field'''.
==Functional form==
==Parameters==
==References==
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66044/ABSTRACT Thomas A. Halgren "Merck molecular force field. I. Basis, form, scope, parameterization, and performance of MMFF94", Journal of Computational Chemistry '''17''' pp.490-519 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66046/ABSTRACT Thomas A. Halgren "Merck molecular force field. II. MMFF94 van der Waals and electrostatic parameters for intermolecular interactions", Journal of Computational Chemistry '''17''' pp. 520 - 552 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66047/ABSTRACT Thomas A. Halgren "Merck molecular force field. III. Molecular geometries and vibrational frequencies for MMFF94", Journal of Computational Chemistry '''17''' pp. 553 - 586 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66048/ABSTRACT Thomas A. Halgren, Robert B. Nachbar "Merck molecular force field. IV. conformational energies and geometries for MMFF94", Journal of Computational Chemistry '''17''' pp. 587 - 615 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66049/ABSTRACT Thomas A. Halgren "Merck molecular force field. V. Extension of MMFF94 using experimental data, additional computational data, and empirical rules", Journal of Computational Chemistry '''17''' pp. 616 - 641 (1996)]
[[category:force fields]]

MMFF94

2010-03-05T12:58:55Z

Nice and Tidy: MMFF94 moved to MMFF94 force field

#REDIRECT [[MMFF94 force field]]

MMFF94 force field

2010-03-05T12:58:55Z

Nice and Tidy: MMFF94 moved to MMFF94 force field

{{Stub-general}}
The '''Merck molecular force field'''.
==References==
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66044/ABSTRACT Thomas A. Halgren "Merck molecular force field. I. Basis, form, scope, parameterization, and performance of MMFF94", Journal of Computational Chemistry '''17''' pp.490-519 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66046/ABSTRACT Thomas A. Halgren "Merck molecular force field. II. MMFF94 van der Waals and electrostatic parameters for intermolecular interactions", Journal of Computational Chemistry '''17''' pp. 520 - 552 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66047/ABSTRACT Thomas A. Halgren "Merck molecular force field. III. Molecular geometries and vibrational frequencies for MMFF94", Journal of Computational Chemistry '''17''' pp. 553 - 586 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66048/ABSTRACT Thomas A. Halgren, Robert B. Nachbar "Merck molecular force field. IV. conformational energies and geometries for MMFF94", Journal of Computational Chemistry '''17''' pp. 587 - 615 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66049/ABSTRACT Thomas A. Halgren "Merck molecular force field. V. Extension of MMFF94 using experimental data, additional computational data, and empirical rules", Journal of Computational Chemistry '''17''' pp. 616 - 641 (1996)]
[[category:force fields]]

MM4 force field

2010-03-05T12:58:35Z

Nice and Tidy:

{{Stub-general}}
The '''MM4''' force field is related to the older [[MM2]] and [[MM3]] [[force fields]].
==Functional form==
==Parameters==
==References==
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66050/ABSTRACT Norman L. Allinger, Kuohsiang Chen, Jenn-Huei Lii "An improved force field (MM4) for saturated hydrocarbons", Journal of Computational Chemistry '''17''' pp. 642-668 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66051/ABSTRACT Neysa Nevins, Kuohsiang Chen, Norman L. Allinger "Molecular mechanics (MM4) calculations on alkenes" Journal of Computational Chemistry '''17''' pp. 669-694 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66052/ABSTRACT Neysa Nevins, Jenn-Huei Lii, Norman L. Allinger "Molecular mechanics (MM4) calculations on conjugated hydrocarbons", Journal of Computational Chemistry '''17''' pp. 695 - 729 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66053/ABSTRACT Neysa Nevins, Norman L. Allinger "Molecular mechanics (MM4) vibrational frequency calculations for alkenes and conjugated hydrocarbons", Journal of Computational Chemistry '''17''' pp. (730 - 746 1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66045/ABSTRACT Norman L. Allinger, Kuohsiang Chen, J. A. Katzenellenbogen, Scott R. Wilson, Gregory M. Anstead "Hyperconjugative effects on carbon - Carbon bond lengths in molecular mechanics (MM4)", Journal of Computational Chemistry '''17''' pp. 747 - 755 (1996)]
[[category:force fields]]

MM4

2010-03-05T12:58:13Z

Nice and Tidy: MM4 moved to MM4 force field

#REDIRECT [[MM4 force field]]

MM4 force field

2010-03-05T12:58:13Z

Nice and Tidy: MM4 moved to MM4 force field

{{Stub-general}}
The '''MM4''' force field is related to the older [[MM2]] and [[MM3]] force fields.
==References==
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66050/ABSTRACT Norman L. Allinger, Kuohsiang Chen, Jenn-Huei Lii "An improved force field (MM4) for saturated hydrocarbons", Journal of Computational Chemistry '''17''' pp. 642-668 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66051/ABSTRACT Neysa Nevins, Kuohsiang Chen, Norman L. Allinger "Molecular mechanics (MM4) calculations on alkenes" Journal of Computational Chemistry '''17''' pp. 669-694 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66052/ABSTRACT Neysa Nevins, Jenn-Huei Lii, Norman L. Allinger "Molecular mechanics (MM4) calculations on conjugated hydrocarbons", Journal of Computational Chemistry '''17''' pp. 695 - 729 (1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66053/ABSTRACT Neysa Nevins, Norman L. Allinger "Molecular mechanics (MM4) vibrational frequency calculations for alkenes and conjugated hydrocarbons", Journal of Computational Chemistry '''17''' pp. (730 - 746 1996)]
#[http://www3.interscience.wiley.com/cgi-bin/abstract/66045/ABSTRACT Norman L. Allinger, Kuohsiang Chen, J. A. Katzenellenbogen, Scott R. Wilson, Gregory M. Anstead "Hyperconjugative effects on carbon - Carbon bond lengths in molecular mechanics (MM4)", Journal of Computational Chemistry '''17''' pp. 747 - 755 (1996)]
[[category:force fields]]