Molecular Dynamcs Smulatons: from ascs to Applcatons. Lecture 7 04.06.01 1
Content of the Lecture 7: 1. Smple thermodynamc averages. Fluctuatons Specfc heat capactes Thermal expanson coeffcent Isothermal compressblty Thermal pressure coeffcent 3. Structural quanttes Par dstrbuton functon Structure factor 4. Tme correlaton functons and transport coeffcents Dffuson coeffcent 5. Structure of bomolecules. Radus of gyraton Secondary structure Intra-molecular hydrogen bonds. 04.06.01
Smple thermodynamc averages. 04.06.01 3
The basc thermodynamc propertes of the system can be calculated n any convenent ensemble. These functons are usually dervatves of one of the characterstc thermodynamc functonsψ ens. The knetc, potental and total nternal energes can be calculated usng the phase functon shown n lecture 1: E = H = K + U (6.1) The knetc energy s a sum of contrbutons from ndvdual partcles momenta, whle the calculaton of the potental energy nvolves summng over all pars, trplets etc. of molecules, dependng on the complexty of the functon. 04.06.01 4
The temperature and the pressure may be calculated usng the vral theorem, whch we wrte here n form of generalzed equpartton : p H p = k T (6. a) So the nstantaneous knetc temperature whose average s equal to T, can be defned as: And the nstantaneous pressure whose average s equal to P, can be defned as follows 04.06.01 q k k H q k k = k T = 1 (6. b) N 1 T = K 3Nk = p m (6.3) 3Nk P + where W s the nternal vral : d ex = ρ kt + W = P P (6.4) N 1 nter W = rf (6.5) 3 = 1 5
Quanttes such as N and can be easly calculated n the smulaton of ensembles n whch these quanttes vary, and derved functons such as enthalpy H, are easly calculated from the above. The drect approach to calculate entropy-related ( statstcal ) propertes such as the Helmholtz (A) and bbs functons (), the chemcal potental (µ) and the entropy (S) t self, s to perform smulaton n grand canoncal ensemble n whch µ or other quantty s specfed. However, as we have already seen, there are some techncal dffcultes assocated wth the smulatons n the grand canoncal ensemble. 04.06.01 6
It should be also stated, that t s dffcult to calculate these propertes n other common ensembles, because they related to the partton functon Q and not to ts dervatves. So we need to calculate Q summng over all the states of the system. It mght seem that to estmate excess statstcal propertes we could use the followng formula: exp( A ex k T ) = Q 1 ex NT = exp ( U kt ) NT (6.6) ut the dstrbuton ρ NT wll be very sharply peaked around the largest values of exp(-u/k T),.e. where exp(u/k T) s comparatvely small. Consequently, any smulaton technque that samples accordng to the equlbrum dstrbuton wll be bound to gve a poor estmate of A by ths route. Specal samplng technques have been developed to evaluate averages of ths type and we wll dscuss them n the next lectures. 04.06.01 7
Fluctuatons. 04.06.01 8
Now, we wll dscuss the nformaton that can be obtaned from the root mean square (RMS) fluctuatons. The most nterestng quanttes are: the constant volume specfc heat capacty C = ( E T ) the constant pressure specfc heat capacty C P = ( H T ) P the thermal expanson coeffcent α P = 1 ( T ) P the sothermal compressblty β T = 1 ( P) T the thermal pressure coeffcent γ = ( P T ) and the adabatc (constant S) analogues of last three. 04.06.01 9
Partally, formulas for these quanttes can be obtaned from the standard theory of fluctuatons Landau and Lfshtz (1980). ut, n computer smulatons we have to be careful to dstngush between properly defned mechancal quanttes such as the energy, or Hamltonan, H, the knetc temperature T or nstantaneous pressure P, and thermodynamc concepts T and P, whch can be only descrbed as the ensemble averages or the parameters defnng the ensemble. Thus, to calculate C n the canoncal ensemble we can smply use a standard formula: σ ( E) = δe = kt C Whereas, snce P s not the same as P, we can not calculate sothermal compressbltyβ T usng the analogous smple formula: ( P) = δp kt βt σ = 04.06.01 10
So we wll start wth the canoncal ensemble. As just mentoned, the specfc heat s gven by fluctuatons n the total energy: δ H = k T C (6.7) NT We can dvde t nto the knetc and the potental contrbutons whch are uncorrelated (.e. δkδu NT =0): NT δ H = δu + δk (6.8) The deal gas part of the specfc heat capacty can be easly calculated from the knetc part, for example for the system of N atoms: 3N d 3 δk = ( kt ) = 3N β C NT = Nk (6.9) And the constant volume specfc heat capacty, can be calculated as follows: δu NT 3 δu = kt ( C 3 Nk ) C NT = + Nk (6.10) k T NT NT 04.06.01 11
Consderaton of the cross-correlaton of the potental energy and the vral fluctuatons wll gve us an expresson for the thermal pressure coeffcentγ : NT ( Nk ) δuδw = k T γ (6.11) And n terms of pressure functon defned n eq. (6.4) we can wrte for a system of N atoms: NT Equaton (6.1) also apples f P s replaced by P or by P ex, whch s more lkely to be avalable n a constant-nt Monte Carlo smulatons. ( γ ρk ) δuδp = k T (6.1) Smlar formulas may be derved for the molecular systems. 04.06.01 1
If we want to consder fluctuatons of the vral tself, we have to defne a further hyper-vral functon: Whch becomes for a parwse addtve potental, It s then easy to show that ( rj r j )( rkl r kl ) X = U (6.13) 1 9 j> k l> k ( r ) where x( r) ( ) dw r X = 1 9 x j = r (6.14) dr NT j> ( 1 Nk T + W β X ) δw = k T + Intermolecular vral or k T Nk T X 1 NT δp = + P βt + (6.16) NT NT 3 Despte the fact the X s a non-thermodynamc quantty, t can be evaluated n computer smulaton, and so eqns. (6.15) and (6.16), provde a route to sothermal compressbltyβ T. NT T NT (6.15) 04.06.01 13
All fluctuaton expressons can be derved for the mcro-canoncal ensemble usng the formula for the transformaton between dfferent ensembles and the fact that the values of smple averages remans unchanged by ths transformaton due to equvalence of ensembles. Specfc heat n mcro-canoncal ensemble may be obtaned from the fluctuatons n the separate potental and knetc components: Cross-correlaton of the pressure fluctuatons and the knetc energy yelds the thermal pressure coeffcent: NE NE ( 1 3Nk C ) 3 δu = δk = Nk T NE NE ( 1 3γ C ) δpδk = δpδu = Nk T (6.17) (6.18) The expresson for fluctuatons of P n the mcro-canoncal ensemble gves the sothermal compressblty: δp 04.06.01 NE = kt NkT 3 + P NE β 1 T + Tγ C + X NE (6.19) 14
Converson from the canoncal to the sothermal-sobarc ensemble can be easly done. Most of formulas of nterest are very smple snce they nvolve well-defned mechancal quanttes. At constant T and P, both volume and energy fluctuatons may occur. The volume fluctuatons are related to the sothermal compressbltyβ T, δ = k NPT The formula for the specfc heat may be obtaned by calculatng the nstantaneous enthalpy, ( H + P ) = kt CP δ (6.1) The thermal expanson coeffcent may be calculated from the cross-correlaton of enthalpy and volume: NPT ( + P ) = kt α P δδ H (6.) NPT Tβ T (6.0) 04.06.01 15
In the grand canoncal ensemble, energy, pressure and number of partcles fluctuate. The number of partcle fluctuatons yeld sothermal compressblty: The formula for specfc heat s obtaned by consderng a functon: δ δn µ T ( H µ N ) = k T Cµ = k T [ ( H µ N ) T ] µ µ T ( N µ ) = ( N ) ktβt = k T (6.3) (6.4) And the usual specfc C (.e. C N ) s obtaned by thermodynamc manpulatons: C = 3 Nk + 1 k T The thermal expanson coeffcent may be derved n the same way T ( ) δu δuδn µ T δn µ T δu δ µ T µ T µ T (6.5) Pβ δuδn N T µ T α P = + (6.6) T Nk T N k T 04.06.01 16
Structural quanttes. 04.06.01 17
The structure of the smple mono-atomc flud can be characterzed by a set of dstrbuton functons for the atomc postons, the smplest of whch s the par dstrbuton functon g (r,r j ), or g (r j ), or smply g(r). Ths functon gves the probablty of fndng a par of atoms a dstance r apart, relatve to the probablty expected for a completely random dstrbuton at the same densty. To defne g(r), we have to ntegrate the confguratonal dstrbuton functon over the postons of all atoms except two, ncorporatng the approprate normalzaton factor. So n the canoncal ensemble: g ( r r ) ( N 1) N, = dr3dr4... dr exp N 1 ρ Z ( βu ( r, r,... r )) 1 N NT (6.7) Obvously the choce of = 1, j = s arbtrary n a system of dentcal atoms. 04.06.01 18
An equvalent defnton takes an ensemble average over pars: g ( r) = ρ δ ( r ) δ ( rj r) = δ ( r rj ) j Ths last form may be used n the evaluaton of g(r) by computer smulaton; n practce, the delta functon s replaced by a functon whch s non-zero n a small range of separatons, and a hstogram s compled of all par separatons fallng wthn each such range. N j (6.8) 04.06.01 19
The par dstrbuton functon s useful, not only because t provdes nsght nto the lqud structure, but also because the ensemble average of any par functon may be expressed n the form: a 1 ( r ) ( ) ( ), r j dr drj g r, rj a r, rj = (6.8) or A = j> a ( r ) = rj Nρ, 1 a( r) g( r) 4 π dr 0 (6.9) The later expresson can be useful f we want to obtan such propertes as energy, pressure of chemcal potental. 04.06.01 0
For example we may wrte the energy (assumng par addtvty) or the pressure although n practce a drect evaluaton of these quanttes, as dscussed above, wll usually gve more accurate results. 04.06.01 ( ) Nk T + Nρ π (6.30) E = 3 r υ( r) g( r) dr P = NkT ρ r w( r) g( r) dr 0 ( 3) N π (6.31) The relaton of chemcal potental to the par dstrbuton functon g(r) can be wrtten n followng way: µ = T k ln ( ) 1 3 ρλ + 4π ξ 0 0 0 r υ( r) g( r; ξ ) dr (6.3) As usual wth the chemcal potental, there s a twst: the formula nvolves a par dstrbuton g(r,ξ) whch depends upon a parameter couplng the two atoms, and t s necessary to ntegrate over ths parameter. 1
The defnton of par dstrbuton functon may be extended to the molecular case when the functon g(r j,ω,ω j ) depends upon the separaton between, and the orentatons of molecules. Exst a number of approaches whch gve partal descrpton of orentatonal orderng: Sectons through g(r j,ω,ω j ) are calculated as a functon of r j for fxed relatve orentatons g(r j,ω,ω j ) can be represented as a sphercal harmonc expanson, where the coeffcents are functons of r j. a set of ste-ste dstrbuton functons g ab (r ab ), can be calculated n the same way as the atomc g(r) for each type of ste. The later case s most smple, however, the number of ndependent g ab (r ab ) functons wll depend on the complexty of the molecule. For example n a three-ste OCS, the sotropc lqud s descrbed by sx ndependent g ab functons (for OO, OC, OS, CS, CC, SS dstances). Although these dstrbuton functons contan less nformaton than g(r j,ω,ω j ), they have the advantage of beng drectly related to the structure factor of the lqud, the expermentally observable propertes. 04.06.01
Now we wll turn to the defnton of quanttes that depend upon the wavevector rather than on poston. In a smulaton wth perodc boundares, we a restrcted to wavevectors that are commensurate wth the perodcty of the system,.e. wth the sze of smulaton box. Specfcally, n a cubc box, we may examne fluctuatons for whch: ( π L )( k, k, k ) k = (6.33) where L s the box length, and k x, k y, k z are ntegers. x y z One quantty of the nterest s the spatal Fourer transform of the number densty: N = 1 ( k ) ( k) = exp r ρ (6.34) 04.06.01 3
Fluctuatons n ρ(k) are related to the structure factor S(k) ( k) ( k) S k = N 1 ( ) ρ ρ (6.35) whch may be measured by neutron or X-ray scatterng experments. Thus S(k) descrbes the Fourer components of densty fluctuatons of the lqud. The structure factor S(k) related, through a three-dmensonal Fourer transform to the par dstrbuton functon = + = + = + sn kr S( k) 1 ρhˆ( k) 1 ρgˆ( k) 1 4πρ r g( r) dr 0 kr where we have ntroduced the Fourer transform of the total correlatons functon h(r) = g(r) 1, and gnored a delta functon contrbuton at k=0. (6.36) 04.06.01 4
Tme-correlaton functons and transport coeffcents. 04.06.01 5
Correlaton between two dfferent quanttes A and are measured n the usual statstcal sense, va the correlaton coeffcent c A c = δaδ σ ( A ) σ ( ) (6.37) A Schwartz nequaltes guarantee that the absolute value of c A les between 0 and 1, wth values close to 1 ndcatng a hgh degree of correlaton. We can evaluate correlaton coeffcent at two dfferent tmes, the resultng quantty s a functon of the tme dfference t, and called tme correlaton functon. The dentcal phase functon c AA (t), called an autocorrelaton functon are of great nterest n computer smulatons because: 04.06.01 They gve a clear pcture of the dynamcs n a flud; Ther tme ntegrals t A may often be related drectly to the mcroscopc transport coeffcents; Ther Fourer transformsĉ AA (ω) may often be related to expermental spectra. 6
We wll gve only few comments to the tme correlaton functons. The non-normalzed correlaton functon s defned ( t) δ( 0) ens = δaγ( t) δ Γ( ) ens C = A 0 A δ (6.38) so that A ( t) C ( t) σ ( A ) σ ( ) c = (6.39) A or ( t) C ( t) ( ) C ( t) ( 0) c = σ = (6.40) AA AA A AA CAA The C A (t) s dfferent for dfferent ensembles, and we can transform t usng a formula for the transformaton between varous ensembles. 04.06.01 7
The computaton of the C A (t) may be thought of as a two step process: 1. Frst, we have to select ntal state ponts Γ(0), accordng to desred dstrbuton ρ ens (Γ), over whch we wll subsequently average.. Second, we must evaluate Γ(t). Ths means solvng the true (Newtonan) equaton of moton. y ths means, tme-dependent propertes may be calculated n any ensemble. In practce, the mechancal equatons of moton are almost always used for both purpose,.e. we use molecular dynamcs to calculate tme correlaton functons n the mcrocanoncal ensemble. Some attenton have to be pad to the queston of the ensemble equvalence, snce the lnk between correlaton functons and transport coeffcents s made through lnear response theory, whch can be carred out n vrtually any ensemble. 04.06.01 8
The transport coeffcents are defned n terms of the response of a system to a perturbaton. For example, the dffuson coeffcent relates the partcle flux to a concentraton gradent, whle a shear vscosty s a measure of the shear stress nduced by an appled velocty gradent. y ntroducng such perturbatons nto the Hamltonan, or drectly nto the equatons of moton, ther effect on the dstrbuton functonρ ens can be calculated. enerally we produce a tme-dependent, non-equlbrum dstrbuton ρ(t) =ρ ens + δρ(t). Hence, any non-equlbrum ensemble average may be calculated. y retanng the lnear terms n the perturbaton, and comparng the equaton for the response wth a macroscopc transport equaton, we may dentfy the transport coeffcent. 04.06.01 9
Ths s usually the nfnte tme ntegral of an equlbrum tme correlaton functon of the followng form: 0 ( ) ( 0) = dt A t A γ (6.41) where γ s the transport coeffcent and A s the varable appearng n the perturbaton term n Hamltonan There s also Ensten relaton, whch s assocated wth any expresson of ths knd ( A ( ) ( 0) ) = t A tγ (6.4) whch holds at large t, compared to the correlaton tme of A. The connectons between eqns. (6.41) and (6.4) can be easly establshed by ntegraton by parts Please keep n mnd, that only a few genune transport coeffcents exst,.e. for only a few hydrodynamc varables A eqns. (6.41) and (6.4) gve a non-zero γ. 04.06.01 30
In computer smulatons, transport coeffcents may be calculated from equlbrum correlaton functons, or usng Ensten relaton. Now we wll dscuss the equatons for calculatng thermal transport coeffcents n the mcro-canoncal ensemble, for a system compose of N dentcal partcles. The dffuson coeffcent D s gven (n three dmenson) by D 1 3 0 ( ) v ( 0) = dt v (6.43) where v (t) s the center-of-mass velocty of a sngle molecule. The correspondng Ensten relaton, vald at a long tmes, s t ( ) r ( ) 1 td 3 r t 0 = (6.44) Please note that n the calculaton of the eqn. (6.44) t s mportant to swtch from one perodc mage to another. 04.06.01 31
Translatonal dynamcs of Na + ons. Tme dependence of the mean-square dsplacement of ons Dependence of long-range on moblty D τ on the hydraton level Γ <r > / A 60 40 0 Γ = 30 D τ / 10-9 m s -1 0.10 0.08 0.06 0.04 I II III I 04.06.01 Γ = 1 0 100 00 300 400 500 600 D τ = t / ps r ( τ + ) r d ( τ ) 0.0 1 14 16 18 0 4 6 8 30 τ = 350 ps = 150 ps d = 3 Γ 3
Structure of bomolecules. 04.06.01 33
Radus of gyraton. Radus of gyraton, R g, s the measure of the sze of the object or an ensemble of ponts. It s calculated as the root mean square dstance of the objects parts from ts center of mass. R g = 1 N N ( r r ) k c. o. m. k = 1 04.06.01 34
T=380 K Rg=6. Å T=340 K T=300 K Rg=8.5 Å T=80 K 04.06.01 35
Secondary structure elements. The structure of protens frequently decomposed nto prmary, secondary, tertary and quaternary structure. There are three man motves n proten secondary structure: α-helx, β-sheet, and random col. However exst a large varety of secondary structure elements, that a not so common and can be found only n partcular protens, among them: 7 -helx, 3 10 helx, π-helx, poly-prolne helx β-turns Secondary structure elements can be descrbed wth help of Ramachandran plots, where torson angles for C -N-C α -C (Φ) and N-C α -C -N (Ψ) are plotted. 04.06.01 36
-10-60 0 60 10-10 -60 0 60 10 10 Β-sheet PPII 10 10 +1 β-turn II 10 ψ 60 0 R 7 60 0 ψ 60 0 + + 60 0-60 -10 Π R 3 R 10 α R -60-10 -60-10 β-turn I +1-60 -10-10 -60 0 60 10 ϕ -10-60 0 60 10 ϕ 04.06.01 37
β-turn type II β-turn type I 04.06.01 38
Intra-molecular hydrogen bonds. A hydrogen bond s the attractve nteracton of a hydrogen atom wth an electronegatve atom, such as ntrogen, oxygen, fluorne or sulfur, that comes from another molecule or chemcal group. A hydrogen atom attached to a relatvely electronegatve atom s a hydrogen bond donor. An electronegatve atom such as fluorne, sulfur, oxygen, or ntrogen s a hydrogen bond acceptor, regardless of whether t s bonded to a hydrogen atom or not. 04.06.01 39
α - helx β - sheet α < 60 N H O C r = 3.5 Å 04.06.01 40
N-termnus C-termnus Ntrogen N P P P N-termnus 4th mnor dagonal A P P P A P P P Oxygen 3rd mnor dagonal nd mnor dagonal N P P P 1st mnor dagonal Man mnor dagonal Man major dagonal C-termnus Ntrogen N P P P Π-helx =5 A P P P A P P P Oxygen α-helx =4 310 -helx =3 N P P P 7-helx = Man mnor dagonal Dsordered structure β-sheet 04.06.01 41
The shear vscosty s gven by: or by Here η = dt Pαβ ( t) Pαβ ( 0) (6.45) k T 0 t ( ) J ( 0) ( J ) η = αβ t αβ kt s an off-dagonal (α β) element of the pressure tensor. + (6.46) 1 P = + αβ pα pβ m r α fβ (6.47) or P αβ 1 = p α p β m j> r jα f jβ (6.48) 04.06.01 4
The 1 J = αβ r α p β (6.49) The negatve of the P αβ s often called the stress tensor. These propertes are mult-partcle, propertes of the system as a whole, and so no addtonal averagng over the N partcles s possble. Consequently η s subjected to much greater statstcal mprecson than D. Some mprovement can be made by averagng over dfferent components,αβ=xy, yz, zx, of P αβ 04.06.01 43
The bulk vscosty s gven by a smlar expresson: η = = 9 k T k 0 T dt αβ 0 δp dt δp αα ( t) δp ( 0) ( t) δp ( 0) ββ = (6.50) where we sum overα,β=x, y, z and note that 1 P = 3 Rotatonal nvarance leads to the equvalent expresson α P αα η 4 + 3η = dt δpαα ( t) δpαα ( 0) (6.51) k T 0 04.06.01 44
The dagonal stresses have to be evaluated wth care, snce a non-vanshng equlbrum average must be subtracted: δp αα ( t) = P ( t) P = P ( t) αα αα αα P (6.5a) δp ( t) = P ( t) P = P ( t) P (6.5b) wth P αα gven by expresson smlar to eqn. (6.48) The correspondng Ensten relaton s: t Pt ( ) ( η 4 η) = ( t) J ( 0 ) + 3 αα αα kt J (6.53) 04.06.01 45
The thermal conductvtyλ T can be wrtten as follows: λ = k T The Ensten relaton s: t ( ) ε t j ( ) ε dt j 0 α α 0 T (6.54) Here, j αε s a component of the energy current, that s the tme dervatve of: δε α = 1 r α ( ) ( ) ( δε ) α δε λ = t 0 α kt T (6.55) ( ε ε ) (6.56) 04.06.01 46
The term r α ε makes no contrbuton f r α = 0 as n a normal one-component MD smulaton. In calculaton of the energy per moleculeε, the potental energy of two molecules (assumng parwse potentals) s taken to be dvded equally between them: ( r ) 1 ε = p m + υ j (6.57) j Please note, the expresson for η and λ T are ensembledependent and the above equatons hold for the mcrocanoncal case only. 04.06.01 47
Transport coeffcents are related to the long-tme behavor of correlaton functons. Short-tme correlatons, on the other hand, may be lnked wth statc equlbrum ensemble averages, by expandng n a Taylor seres. For example, the velocty of partcle may be wrtten v ( t) = v ( 0) + v ( 0) t + 1 v ( 0) t +... Multplyng by v (0) and ensemble averagng gves us: v ( t) v ( 0) = v + 1 v v t +... = (6.58) = v 1 v t (6.59) Thus the short-tme velocty auto-correlaton functon s related to the mean square acceleraton,.e. to the mean square force. 04.06.01 48
Ths behavor can be used to defne the Ensten frequencyω E ( ) ( ) ( 0 1...) 1 t v = v ω t + v (6.60) The analogy wth Ensten model, of an atom vbratng n the mean feld of ts neghbors, wth frequency ω E n the harmonc approxmaton becomes clear when we replace the mean square force by the average potental curvature usng E f α And the result s: = fα U r α = kt U r α (6.61) E = m f α v = 1 3m r ω (6.6) U Ths may be easly evaluated n computer smulaton for a parwse potental. 04.06.01 49
There are some other correlaton functons of nterest n computer smulatons whch we wll dscuss here very brefly. The generalzaton of eqn. (6.35) to the tme doman gves the ntermedate scatterng functon I(k,t) I 1 ( k, t) = N ( k, t) ρ( k,0) ρ (6.63) The temporal Fourer transform of ths functon, the dynamc structure factor S(k,ω), may be measured by nelastc neutron scatterng. Spatally Fourer-transformng I(k,t) yelds the van Hove functon (r,t), a generalzaton of g(r) whch measures the probablty of fndng a partcle at poston r at tme t, gven that a partcle was at the orgn of coordnates at tme 0. All these functons may be dvded nto parts due to self (.e. sngle-partcle) moton and due to dstnct (.e. collectve) effects. 04.06.01 50
For a system of rgd molecules, the angular velocty ω plays a role n reorentatonal dynamcs analogous to that of v n translaton. The angular velocty correlaton functon ω(t) ω(0) may be used to descrbe rotaton. Tme-dependent orentatonal correlatons may be defned as straghtforward generalzatons of the quanttes seen earler. For a lnear molecule, the tme correlaton functon of rank-l sphercal harmoncs s: c l * ( t) = 4π Y ( Ω ( t) ) Y ( Ω ( 0) ) = P ( cosδγ ( t) ) lm lm l (6.64) whereδγ(t) s the magntude of the angle turned through n tme t. Note that there are l+1 rank-l functons, all dentcal n form, correspondng to dfferent values of m. Please note, that the frst-rank auto-correlaton functons may be related to nfra-red absorpton, and second-rank functons to lght scatterng. Functons of all ranks contrbute to nelastc neutron scatterng spectra from molecular lquds. 04.06.01 51