Carlo Adamo. Density Functional Theory
|
|
- Nora Joseph
- 5 years ago
- Views:
Transcription
1 Carlo Adamo Densty Functonal Theory Concepts and models Équpe de Modélsaton des Systèmes Complexes Laboratore d Électrochme et Chme Analytque CNRS, UMR 7575 cole Natonale Supéreure de Chme de Pars 1 The ngredents (1): Densty : It provdes us nformaton about how somethng s dstrbuted/spread on a gven space (volume) lectron Densty : It tells us where the electrons are lkely l to exst * ( ( ( ( r ) lectron densty s an «observable» exp bs(dmnosuccnontrlo)nckel DFT 1
2 The ngredents (): A functon (f) maps a set of numbers to another set of numbers Y=f(X) X Y=X Y A functonal (F) s a functon of a functon A functon whch maps a set of functons to a set of numbers x. F[A(X),B(X),C(X),.] xample : F xc [(,(] X A(X) Y B(X) 013 C(X) D(X) Y 3 Why DFT? From a pragmatc pont of vew:. Brght Wlson, 1965 The knowledge of the densty s all that t s necessary for a complete determnaton of all molecular propertes. If one knows the exact electron densty, (, then the cusps of ths densty would occur at the postons of the nucle. Furthermore, a knowledge of ( at the nucle would gve ther nuclear charges. Thus the full Schrödnger Hamltonan was known because t s completely defned once the poston and charge of the nucle are gven. In prncple, the wavefuncton and are known, and thus everythng s known. 4
3 Can the total energy be expressed as a functon of the densty? Hˆ Hˆ ( r1, r1 ) 1 ( Vext ( ( r1, r ) 1 ' r r1 r1 1 ' 1 ( r1, r1 ) N... ( r1 1, x...x N ) * ( r1 1, x... x N ) d 1dx... dx N N( N 1) ( r1, r )... ( x1, x... x N ) d 1d dx3... dx N Crtcal ngredents: knetc energy and electron-electron nteracton 5 arly attempts: Thomas-Ferm (197) Hˆ T [ ] ( V ( [ ] ext ee 1) xact kn. en. (T[]) s substtuted by the kn. en. of a homogenous electrons gas ) The external potental (V ext []) s that generated by the nucle 3) The electron-electron nteracton s the Coulomb repulson TF T [ ] ( v( [ ] TF ee 5/ 3 c f c F =constante de Ferm rrd Nucle-electrons nteracton (Coulomb) 1 Z A r A r R A e - /e - (Coulomb) ( r1 ) ( r r 1 ) 1 J[ ] 6 3
4 Thomas-Ferm (197): how does t works? Not very well : n TF theory no molecular system s stable relatve to dssocaton nto constutents fragments non bondng theorem (Teller (196) / Balazs (1967) / Leb (1973) / Smon (1977)) TF T Z A 1 ( r1 ) ( r ) [ ] ( 1 r r r TF A R A 1 What s mssng? xchange Correlaton and furthermore T TF s local classcal Coulomb nteractons 7 Add exchange : Thomas-Ferm-Drac model (TFD) TF TF T c TF f [ ] ( ( v( J[ ] K[ ] 5/3 ( v( J[ ] c x ( c f = 3/10(3 ) / 3 c x = 3/4(3/) 1/ 3 4/3 Add gradent correctons to knetc energy : Thomas-Ferm-Drac-Wezsacker model (TFDW or TFD-W) TF c f 5/3 1 ( ( r ) r v r J cx r r ( ) ( ) [ ] ( ) 8 ( ) 4/3 verythng s expressed as a functon of the densty (or gradent of the densty) 8 4
5 A dfferent approach: Slater and the X model (or HFS model) HF SLATR 1 Z J (1) K (1) (1) (1) A 1 a a A r R A a1, N / 1 Z A V (1) (1) (1) (1) 1 coulomb VXHF A r RA C x = constant 075< 0.75 < 1 V XHF c x r 1/ 3 xchange only; no correlaton Works XTRMLY well (and t s stll used especally for sold state) 9 X performances 10 5
6 The Hohenberg and Kohn theorems (1964) v ext ( r ) 0 r ( ( 0 All propertes of the many-body system are determned by the ground state densty 0 ( ach property s a functonal of the ground state densty 0 ( whch s wrtten as f [ 0 ] 11 Hohenberg and Kohn: theorem I r 0 ( defne the external potental V ext ( and thus all propertes (except for constant) 1 6
7 r r r r 13 r r In prncple, one can fnd all other propertes and they are functonals of 0 (. HK F T V V r HK V ext nt r r r d ext 14 7
8 QC v ext ( r ( ( DFT- HK 0 0 All observables How to get the wf from the densty? How do I know, gven an arbtrary functon (, that t s a densty comng from an antsymmetrc N-body wavefuncton Ψ(r 1,...,r N )? N-representablty Solved: any square-ntegrable nonnegatve functon satsfes t There s an nfnte number of w.f. yeldng How to get the ground state densty (? How do I know, gven an arbtrary functon (, that t s the ground state densty of a local potental v(? V-representablty Constraned-search formalsm (Levy-Leb) Defnton of the unversal functonal 15 Constraned search formalsm (Levy and Leb (1977)) Double mnmzaton procedure: 0 mn mn Tˆ Vˆ ext Vˆ ee 1 Wavefunctons gvng densty ( r ) Tˆ Vˆ 0 mn mn ee ( r ) V F mn Tˆ ˆ V ee ext Defnton of the unversal functonal : ˆ ˆ F mn T V ee 16 8
9 Double mnmsaton or how to fnd the tallest chld n a school? 1. Fnd the tallest n each classroom mn. Fnd the tallest of the tallest mn 17 From Parr&Yang Varous extensons have been proposed: xtensons to spn dependent systems (Barth, Hedn, 197) [ n, n ] xtenson to relatvstc systems (Vgnale, Kohn, 1988) [ jr ( )] xtenson to fnte temperatures Fn [ ] n [ ] TSn [ ] Tme-Dependent DFT (Runge, Gross, 1984) 18 9
10 What have we ganed so far? Apparently Nothng: The only result s that the densty determnes the potental We are stll left wth the orgnal many-body problem If you don t lke the answer, change the queston Kohn and Sham ansatz (1965) Replace the nteractng-partcles hamltonan wth one that t can be solved more easly KS hamltonan: an hamltonan descrbng N non-nteractng partcles assumed to have the same densty as the true nteractng system. Hˆ KS 1 veff r 1/ det... N! 1 3 N 19 Kohn and Sham ansatz vsualzed Hˆ 1 KS v eff r As f non-nteractng electrons n an effectve (self-consstent) potental From. Artacho 0 10
11 Let us start agan wth the defnton of the energy : HK HK TVnt Vext ( ( xpress the energy usng the KS ansatz TS J XC Vext ( ( And get the defnton of exchange correlaton energy : XC HK TS J TT V J S ee Knetc All non classcal contrbutons (.e. non Coulomb, J) to electron-electron nteracton 1 Hˆ HK KS TS J XC Vext ( ( 1 ˆ KS H Hˆ KS v eff r 1 ( r' ) ' vxc r r r' r vext 1 v Hartree r r, 1, N, r v xc r v ext r HF lke equatons v xc xc r 11
12 Intal guess: and Calculate the effectve potental (v eff ) r v r v r v r How does t works n practse v Smlartes wth HF: eff Hartree xc ext A bass set s stll needed, but can be more flexble (numercal bass functons) Soluton of secular equaton Solve KS equatons (v eff ) SCF procedure s stll used Hˆ 1 KS v eff r Calculate electron densty ( f Dfferences wth HF: e - correlaton s mplctly ncluded The soluton of the secular equaton s computatonally more effcent formally scales as N 3 as opposed to N 4. No Yes Happy end Test convergence Output quanttes 3 Advantages of KS equatons All dffcult terms to be computed are collected n xc If v xc s exact the KS soluton s exact - exchange s ncluded - correlaton s ncluded - knetc energy s computed from orbtals (contrary to TF theory) - scalng s O(N 3 ) (1). The wave functon of an N-electron system ncludes 3N varables, whle the densty, no matter how large the system s, has only three varables x, y, and z. Movngfrom[] to[] n computatonal chemstry sgnfcantly reduces the computatonal effort needed to understand electronc propertes of atoms, molecules, and solds. (). Formulaton along ths lne provdes the possblty of the lnear scalng algorthm, whose computatonal complexty goes lke O(NlogN), essentally lnear n N when N s very large. (3). The other advantage of DFT s that t provdes some chemcally mportant concepts, such as electronegatvty (chemcal potental), hardness (softness), Fuku functon, response functon, etc.. 4 1
13 Meanng of KS genvalues ( f ( f= occupaton number (could be fractonal) The onzaton potental s I k N 1 k N k 1 v d fk k f k fk d fk Usng the mean-value approxmaton I 0.5 k k Ths s the so-called Janak-(Slate transton state theorem It could be consdered as the equvalent of the Koopman s theorem 5 The problem: v xc exact s unknown Dfferent approxmate forms of vxc have been proposed. The theory s exact, the functonals are approxmate 6 13
14 Consequences of the use of approxmate v xc : self nteracton error (SI) TS J XC Vext ( ( HK TS J T T V J HK XC HF H 1 1,N S, j1,n ee In exact KS as n HF: there s no Coulomb nteracton of one electron wth tself SI(N) J j K j But n approxmate DFT ths s not the case: J xc,0 7 ffects of self nteracton error He + BLYP LDA (He + +He) He+ PW91 CCSD(T) (He +0.5 ) 8 14
15 How to get an approxmate xc functonal? Contans nformaton on the many-body system of nteractng electrons The easest way: Local Densty Approxmaton LDA Assume the functonal s the same as a model problem the homogeneous electron gas Separate the exchange and correlaton contrbutons: xc = x + c xc can be calculated as a functon of the densty only 9 LDA model problem : the homogeneous electron gas LDA XC [ ] ( XC ( ( ) ( X ( ( ) C ( ( ) probablty of fndng the partcle at r homogeneous electron gas exchangecorrelaton energy per partcle (at the pont The value of the xc energy depends only on the local densty. The e- densty () may vary as a functon of r, but s sngle-valued, and the fluctuatons n away from r do not affect the value of xc at r
16 Is the Local Densty Approxmaton physcally soundng? around each electron other electrons tend to be excluded Defnton of x-c hole : ( r, r') xc xcs the nteracton of the electron wth the hole : t nvolves only a sphercal average ( r,r' ) [ ] r xc ' Sphercal average xc r - r' around electron xchange hole n Ne atom Gunnarsson, et. al. nucleus Very non-sphercal! electron Sphercal average very close to the hole n a homogeneous electron gas! 50 n x (r, r ) xact r r r r r r o = LD (r /o 0 n x(r, r ) + xact r = 0.4 o 0 LD (r /o xact xact LD + LD r n x 5.0 (r, r ) r r r = 0.09 o (a) r /o n r n x 3.0 (r, r ) r = 0.4 o (b) r xchange-correlaton (x-c) hole n slcon Calculated by Monte Carlo methods xchange Correlaton (a) (b) Hole s reasonably well localzed near the electron Supports a local approxmaton Hood et al 3 16
17 LDA model problem : the homogeneous electron gas LDA XC [ ] ( XC ( ( ) ( X ( ( ) C ( ( ) Get an expresson for them 33 LDA : xchange part LDA C x( X (per partcle) 1/ 3 Derved by Bloch et Drac (199/1930) for homogeneous electron gas Functonal form dentcal to that of Slater (HFS) Usually called Slater exchange functonal LDA X [ ] ( X ( ( ) LDA : Correlaton part No explct formulaton Approxmate analytcal expresson to reproduce accurate quantum Monte-Carlo (Ceperly & Alder, 1980) results for a homogeneous electron gas Most used LDA approxmaton for correlaton Volsko, Wlk et Nusar (1980): VWN. LDA A x b Q bx0 ( b x0 ) Q C ln arctan ln( x x0 ) arctan X ( x) Q x b X ( x0 ) Q x b34 17
18 LDA: how t works v xc Dscontnuty of the potental for the fllng of a electronc shell xc r xc xc r From A. V. Morozov 35 LDA: how t works A: Hgh densty, large knetc energy, LDA approxmaton unmportant B: Small densty gradent, LDA s good C:large gradent, LDA fals 36 18
19 Open shell systems and Local Spn Densty Approxmaton (LSDA) Dfferent denstes for dfferent spns : splt the total densty ( ( ( LSDA XC [, ] ( XC ( (, ( ) Measure of spn polarzaton : ( r ) ( r ) ( Remark : n prncple, snce the external potental s spn ndependent there s no need to splt the dfferent spn denstes 37 How to amelorate the LDA? Atoms, molecules or solds are not a homogeneous electron gas : Include non local effects GA (Gradent xpanson Approxmaton): F(( GA XC [, ] ( XC (, ), ' Taylor expanson C, XC (, ) / 3 / 3... Note: mathematcally speakng GAs are stll local How do they work? Not a great mprovement Reason : xchange correlaton hole propertes not satsfed 38 19
20 GGA (Generalzed Gradent Approxmaton) Impose the fulfllment of the propertes of the exchange-correlaton hole GGA XC [, ] f (,,, ) GGA XC GGA X GGA C GGA LDA X X F( s ) ( where s s the reduced densty Measure local nhomogenty s 4 / 3 ( 4 / 3 ( r ) ( 3 S hgh for hgh gradent or small densty regons (far from nucle) S small for small gradents (bondng regon) S ntermedate for hgh gradent and small densty (near the nucle) 39 xample of commonly used exchange functonals Becke, 1988 (B ou B88) s F B s snh s Perdew, 1986 (P ou P86) 86 F P s (4 ) 1/ 3 s 14 (4 ) 1/ 3 4 s 0. (4 ) 1/ 3 6 1/15 Functonal form can get extremely complex (especally for correlaton functonals) Problem: How to get a GGA? 40 0
21 Parametrzed functonals sem-emprcaluse adjusted parameters to reproduced exact or expermental data (ex. atomc energes) Over parametrsaton Sem-emprcal Good propertes Non parametrzed functonals mpose physcal constrans (unform electron gas lmt, asymptotc behavo «unversal» (worse) chemcal propertes Bref (and extremely non exhaustve) lst of commonly used GGA functonals F authors x F authors c B PW91 PB mpw HCTH B97 Becke (1988) Perdew et Wang (1991) Perdew, Burke, rnzerhof (1996) Adamo, Barone (1997) Handy et al. (1999) Becke P86 LYP PW91 PB Perdew (1986) Parr et al. (1988) Perdew et Wang (1991) Perdew, Burke, rnzerhof (1996) 41 Some theoretcal contrants Sze consstency : (AB)=(A)+(B) Vral Theorem Self-nteracton error Janak Theorem Leb-Oxfod bound Coordnate scalng Hrao JCP
22 Leb-Oxford Lmt Bondng regon Adamo JCP Atomsaton energes (n ev) of several molecules: theory vs experment. HF LDA GGA xp. H H O HF O F CH Solds : damond Property HF LDA GGA xp. a 0, Å a, ev K 0, GPa
23 How to mprove GGA results? HAVN (chemcal accuracy) rung 5 rung 4 rung 3 rung + explct dependence on unoccuped orbtals + explct dependence on occuped orbtals + explct dependence on knetc energy densty +explct dependence on gradents of the densty John Perdew Jacob Ladder* fully nonlocal hybrd functonals meta-ggas GGAs rung 1 local densty only LDA ARTH (Hartree theory) *DFT conference Menton 000 Update 007: Rung 4bs: Hyper GGAs 45 Phlppe Ratner meta-gga or dependent F xc 1, occ knetc energy densty Introduce quas-local nformaton 46 3
24 How to mprove the exchange energy : Hybrd Functonals Idea: HF exchange s exact. Therefore we could thnk to combne exact (HF) exchange wth a (GGA ) correlaton functonal (Le & Clement 1974) XC exact X KS C How does t work? Very bad! Mean average error 3 kcal/mole for the G ensemble (50 molecules) whle a MA of 5-7 kcal/mol s obtaned wth standard GGA Actually combnng a percentage of HF exchange wth a GGA exchange works much better!!! Theoretcal justfcaton : adabatc connecton (Becke 1993 Half and Half) In practse : HF% between 0 and 30% 47 B3LYP B3 xc a x0 LSD x (1 a xo ) HF x a x1 B x LSD c a c PW 91 c (Becke, JCP 1993) 3 parameters ftted on G (onzaton and atomzaton energes) PB0 PB0 xc 1 4 HF x 3 4 (Adamo, Scusera JCP 1999) No ftted parameters PB x PB c Normally hybrd functonals outperform all GGA and meta-gga functonals and they are the reference for chemcal applcatons 48 4
25 Method Dstance (Å) D 0 (kcal/mol) Dpole moment (D) Harmonc freq. (cm -1 ) HF & post-hf HF MP CCSD[T] Performance of selected functonals datomc molecules LDA & GGA LSDA BPW BLYP LGLYP PWPW mpwpw Hybrd 3 parameters B3LYP B3PW mpw3pw Hybrd ab-nto B1LYP B1PW LG1LYP Performance of selected functonals MA for harmonc frequences (cm -1 ) (G set,>50 organc molecules) Level of thoery rrorr HF and post-hf HF/6-311G(3df,p) 144 MP/6-31G(d,p) 99 CCSD/6-311G(3df,p) 31 LSDA SVWN/6-31G(d,p) 75 GGA BLYP/6-311G(d,p) 59 BPW91/6-311G(d,p) 69 PWPW91/6-311G(d,p) 66 mpwpw91/6-311g(d,p) 66 Hybrd functonals B1LYP/6-311G(d,p) 33 B1PW91/6-311G(d,p) 48 mpw1pw91/6-311g(d,p) 39 B3LYP/6-311G(d,p) 31 B3PW91/6-311G(d,p) 45 mpw3pw91/6-311g(d,p)
26 Rutle - GTO 51 The XC functonal can be crucal for chemcal understandng Homogeneous catalyss of ethylene R growng chan R 1 N N Al R TS 13 P14 R The catalyst termnaton R 1 = H, R = so-propyl p py ; R 1 = tert-butyl, R = soprop R=growng chan BHT TS 15 G. Talarco, P. H. M. Budzelaar, V. Barone and C. Adamo Chem. Phys. Lett. 39, 99, (000). G. Talarco, V. Barone, P. H. M. Budzelaar and C. Adamo, J. Phys. Chem. A, 105 (001) 9014 G. Talarco, V. Barone, L. Joubert and C. Adamo Int. J. Quantum Chem. 91 (003)
27 functonal f best est. MP VSXC B1Bc95 PB0 B98 mpw0 B1LYP B3PW91 B3LYP BP86 BLYP termnaton channel Dfferent functonals dfferent CHMICAL answers. best est. MP VSXC B1Bc95 PB0 B98 mpw0 B1LYP B3PW91 # ns B3LYP BP86 BLYP fu unctonal nserton channel (kcal/mol) # BHT (kcal/mol) BP86 termnaton most probable (low m.w.) VSXC nserton most probable (hgher m.w.) 53 Known breakdowns of DFT Many efforts to assess the relablty of DFT by a tral-and-error approach among others reacton barrers, CT complexes, Rydberg exctatons, IP from Koopmans, band gap, bond length alternaton, magnetc propertes, p vdw nteractons, (Un)Known causes? Localzaton Self Interacton rror (SI) Non-dynamc correlaton effects All problems comng from the approxmate nature of the XC contrbuton The theory s exact, the functonals are approxmate 54 7
28 Challenges n DFT Better functonals (e.g. CR, MR-05) rror correctons (e.g. SIC, ad-hoc parametrzaton) Statc correlaton (e.g. MC-DFT, RSH) Localzaton vs delocalzaton In chemcal language 55 8
Introduction to Density Functional Theory. Jeremie Zaffran 2 nd year-msc. (Nanochemistry)
Introducton to Densty Functonal Theory Jereme Zaffran nd year-msc. (anochemstry) A- Hartree appromatons Born- Oppenhemer appromaton H H H e The goal of computatonal chemstry H e??? Let s remnd H e T e
More informationProbabilistic method to determine electron correlation energy
Probablstc method to determne electron elaton energy T.R.S. Prasanna Department of Metallurgcal Engneerng and Materals Scence Indan Insttute of Technology, Bombay Mumba 400076 Inda A new method to determne
More informationLecture 14: Forces and Stresses
The Nuts and Bolts of Frst-Prncples Smulaton Lecture 14: Forces and Stresses Durham, 6th-13th December 2001 CASTEP Developers Group wth support from the ESF ψ k Network Overvew of Lecture Why bother? Theoretcal
More informationRobert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations
Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton
More informationThe GW approximation in 90 minutes or so. F. Bruneval Service de Recherches de Métallurgie Physique CEA, DEN
The GW approxmaton n 90 mnutes or so Servce de Recherches de Métallurge Physque CEA, DEN DFT tutoral, Lyon december 2012 Outlne I. Standard DFT suffers from the band gap problem II. Introducton of the
More informationThis chapter illustrates the idea that all properties of the homogeneous electron gas (HEG) can be calculated from electron density.
1 Unform Electron Gas Ths chapter llustrates the dea that all propertes of the homogeneous electron gas (HEG) can be calculated from electron densty. Intutve Representaton of Densty Electron densty n s
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More information5.62 Physical Chemistry II Spring 2008
MIT OpenCourseWare http://ocw.mt.edu 5.62 Physcal Chemstry II Sprng 2008 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 5.62 Sprng 2008 Lecture 34 Page Transton
More information5. Response properties in ab initio schemes
5. Response propertes n ab nto schemes A number of mportant physcal observables s expressed va dervatves of total energy (or free energy) E. Examples are: E R 2 E R a R b forces on the nucle; crtcal ponts
More informationAdvanced Quantum Chemistry III: Part 3. Haruyuki Nakano. Kyushu University
Advanced Quantum Chemistry III: Part 3 Haruyuki Nakano Kyushu University 2013 Winter Term 1. Hartree-Fock theory Density Functional Theory 2. Hohenberg-Kohn theorem 3. Kohn-Sham method 4. Exchange-correlation
More informationMultiscale Modeling UCLA Prof. N. Ghoniem
Interdscplnary graduate course Fall 003 Multscale Modelng UCLA Prof. N. Ghonem Introducton n Electronc Structure Calculatons Ncholas Kousss Department of Physcs Calforna State Unversty Northrdge The Fundamental
More informationMolecular Dynamics and Density Functional Theory
Molecular Dynamcs and Densty Functonal Theory What do we need? An account n pemfc cluster: Host name: pemfc.chem.sfu.ca I wll take care of that. Ths can be usually a common account for all of you but please
More informationwhere the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt
Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.
More informationSUPPLEMENTARY INFORMATION
do: 0.08/nature09 I. Resonant absorpton of XUV pulses n Kr + usng the reduced densty matrx approach The quantum beats nvestgated n ths paper are the result of nterference between two exctaton paths of
More informationElectronic Quantum Monte Carlo Calculations of Energies and Atomic Forces for Diatomic and Polyatomic Molecules
RESERVE HIS SPACE Electronc Quantum Monte Carlo Calculatons of Energes and Atomc Forces for Datomc and Polyatomc Molecules Myung Won Lee 1, Massmo Mella 2, and Andrew M. Rappe 1,* 1 he Maknen heoretcal
More informationQuantum states of deuterons in palladium
Tsuchda K. Quantum states of deuterons n palladum. n Tenth Internatonal Conference on Cold Fuson. 003. Cambrdge MA: LENR-CANR.org. Ths paper was presented at the 10th Internatonal Conference on Cold Fuson.
More information5.76 Lecture #5 2/07/94 Page 1 of 10 pages. Lecture #5: Atoms: 1e and Alkali. centrifugal term ( +1)
5.76 Lecture #5 /07/94 Page 1 of 10 pages 1e Atoms: H, H + e, L +, etc. coupled and uncoupled bass sets Lecture #5: Atoms: 1e and Alkal centrfugal term (+1) r radal Schrödnger Equaton spn-orbt l s r 3
More informationMulti-electron atoms (11) 2010 update Extend the H-atom picture to more than 1 electron: H-atom sol'n use for N-elect., assume product wavefct.
Mult-electron atoms (11) 2010 update Extend the H-atom pcture to more than 1 electron: VII 33 H-atom sol'n use for -elect., assume product wavefct. n ψ = φn l m where: ψ mult electron w/fct φ n l m one
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationElectron-Impact Double Ionization of the H 2
I R A P 6(), Dec. 5, pp. 9- Electron-Impact Double Ionzaton of the H olecule Internatonal Scence Press ISSN: 9-59 Electron-Impact Double Ionzaton of the H olecule. S. PINDZOLA AND J. COLGAN Department
More informationElectronic Structure for Excited States (multiconfigurational methods) Spiridoula Matsika
Electronc Structure for Excted States (multconfguratonal methods) Sprdoula Matska Excted Electronc States Theoretcal treatment of excted states s needed for: UV/Vs electronc spectroscopy Photochemstry
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015
Lecture 2. 1/07/15-1/09/15 Unversty of Washngton Department of Chemstry Chemstry 453 Wnter Quarter 2015 We are not talkng about truth. We are talkng about somethng that seems lke truth. The truth we want
More informationComplex Atoms; The Exclusion Principle and the Periodic System
Complex Atoms; The Excluson Prncple and the Perodc System In order to understand the electron dstrbutons n atoms, another prncple s needed. Ths s the Paul excluson prncple: No two electrons n an atom can
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationIntermolecular force fields and how they can be determined
Intermolecular force felds and how they can be determned Ad van der Avord Unversty of Njmegen Han-sur-Lesse, December 23 p.1 Equaton of state (Van der Waals) of non-deal gas ( p + a )( ) V 2 V b = kt repulson
More informationψ ij has the eigenvalue
Moller Plesset Perturbaton Theory In Moller-Plesset (MP) perturbaton theory one taes the unperturbed Hamltonan for an atom or molecule as the sum of the one partcle Foc operators H F() where the egenfunctons
More informationProf. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model
EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several
More information5.03, Inorganic Chemistry Prof. Daniel G. Nocera Lecture 2 May 11: Ligand Field Theory
5.03, Inorganc Chemstry Prof. Danel G. Nocera Lecture May : Lgand Feld Theory The lgand feld problem s defned by the followng Hamltonan, h p Η = wth E n = KE = where = m m x y z h m Ze r hydrogen atom
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationRate of Absorption and Stimulated Emission
MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld
More informationCI/CEPA. Introduction CI Size Consistency Derivation CEPA EPV Results Remarks
CI/CEPA Introducton CI Sze Consstency Dervaton CEPA EPV Results Remarks 1 HY=EY H e Ψ e = E e Ψe Expanson of the Many-Electron Wave Functon Ψ HF Ψ e = Φ o + a a C Φ + ab ab C j Φ j +... Φo = A φ 1 φ 2...
More informationSupporting information.
Response to Comment on the paper "Restrcted Geometry Optmzaton: A Dfferent Way to Estmate Stablzaton Energes for Aromatc Molecules of Varous Types" Zhong-Heng Yu* and Peng Bao Supportng nformaton. Contents:
More information5. THE ADIABATIC APPROXIMATION
5. THE ADIABATIC APPROXIMATION In quantum mechancs, the adabatc approxmaton refers to those solutons to the Schrödnger equaton that make use of a tme-scale separaton between fast and slow degrees of freedom,
More informationSolutions to Problems Fundamentals of Condensed Matter Physics
Solutons to Problems Fundamentals of Condensed Matter Physcs Marvn L. Cohen Unversty of Calforna, Berkeley Steven G. Loue Unversty of Calforna, Berkeley c Cambrdge Unversty Press 016 1 Acknowledgement
More informationKey Concepts, Methods and Machinery - lecture 2 -
B. Roos V. A. Fok C. C. J. Roothaan W. Hetler Quantum Mechancs W. Paul J. Čížek P.A.M. Drac Key Concepts, Methods and Machnery - lecture - J. A. Pople M. S. Plesset N. Bohr E. Hückel E. Schrödnger and
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More informationDensity matrix. c α (t)φ α (q)
Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationis the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors
Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson
More informationFrequency dependence of the permittivity
Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but
More informationSTATISTICAL MECHANICAL ENSEMBLES 1 MICROSCOPIC AND MACROSCOPIC VARIABLES PHASE SPACE ENSEMBLES. CHE 524 A. Panagiotopoulos 1
CHE 54 A. Panagotopoulos STATSTCAL MECHACAL ESEMBLES MCROSCOPC AD MACROSCOPC ARABLES The central queston n Statstcal Mechancs can be phrased as follows: f partcles (atoms, molecules, electrons, nucle,
More informationDensity Func,onal Theory (Chapter 6, Jensen)
Chem 580: DFT Density Func,onal Theory (Chapter 6, Jensen) Hohenberg- Kohn Theorem (Phys. Rev., 136,B864 (1964)): For molecules with a non degenerate ground state, the ground state molecular energy and
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationSTABILITY OF METALLIC FERROMAGNETISM: CORRELATED HOPPING OF ELECTRONS IN Mn 4 N
STABILITY OF METALLIC FERROMAGNETISM: CORRELATED HOPPING OF ELECTRONS IN Mn 4 N EUGEN BIRSAN 1, COSMIN CANDIN 2 1 Physcs Department, Unversty Lucan Blaga, Dr. I. Ratu str., No. 5 7, 550024, Sbu, Romana,
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationPHY688, Statistical Mechanics
Department of Physcs & Astronomy 449 ESS Bldg. Stony Brook Unversty January 31, 2017 Nuclear Astrophyscs James.Lattmer@Stonybrook.edu Thermodynamcs Internal Energy Densty and Frst Law: ε = E V = Ts P +
More informationInternational Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sc. Technol., 4() (03), pp. 5-30 Internatonal Journal of Pure and Appled Scences and Technology ISSN 9-607 Avalable onlne at www.jopaasat.n Research Paper Schrödnger State Space Matrx
More informationNeutral-Current Neutrino-Nucleus Inelastic Reactions for Core Collapse Supernovae
Neutral-Current Neutrno-Nucleus Inelastc Reactons for Core Collapse Supernovae A. Juodagalvs Teornės Fzkos r Astronomjos Insttutas, Lthuana E-mal: andrusj@tpa.lt J. M. Sampao Centro de Físca Nuclear da
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationESI-3D: Electron Sharing Indexes Program for 3D Molecular Space Partition
ESI-3D: Electron Sharng Indexes Program for 3D Molecular Space Partton Insttute of Computatonal Chemstry (Grona), 006. Report bugs to Eduard Matto: eduard@qc.udg.es or ematto@gmal.com The Electron Sharng
More information5.04, Principles of Inorganic Chemistry II MIT Department of Chemistry Lecture 32: Vibrational Spectroscopy and the IR
5.0, Prncples of Inorganc Chemstry II MIT Department of Chemstry Lecture 3: Vbratonal Spectroscopy and the IR Vbratonal spectroscopy s confned to the 00-5000 cm - spectral regon. The absorpton of a photon
More informationThe GW Approximation. Lucia Reining, Fabien Bruneval
, Faben Bruneval Laboratore des Soldes Irradés Ecole Polytechnque, Palaseau - France European Theoretcal Spectroscopy Faclty (ETSF) Belfast, June 2007 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationDensity Functional Theory I
Densty Functonal Theoy I cholas M. Hason Depatment of Chemsty Impeal College Lonon & Computatonal Mateals Scence Daesbuy Laboatoy ncholas.hason@c.ac.uk Densty Functonal Theoy I The Many Electon Schönge
More informationNote on the Electron EDM
Note on the Electron EDM W R Johnson October 25, 2002 Abstract Ths s a note on the setup of an electron EDM calculaton and Schff s Theorem 1 Basc Relatons The well-known relatvstc nteracton of the electron
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationA critical re-assignment of the Rydberg states of iodomethane based on new polarization data J. Chem. Phys. 138, (2013); /1.
Recent development of self-nteracton-free tme-dependent densty-functonal theory for nonperturbatve treatment of atomc and molecular multphoton processes n ntense laser felds Shh-I Chu Ctaton: The Journal
More informationA quote of the week (or camel of the week): There is no expedience to which a man will not go to avoid the labor of thinking. Thomas A.
A quote of the week (or camel of the week): here s no expedence to whch a man wll not go to avod the labor of thnkng. homas A. Edson Hess law. Algorthm S Select a reacton, possbly contanng specfc compounds
More informationThermodynamics and statistical mechanics in materials modelling II
Course MP3 Lecture 8/11/006 (JAE) Course MP3 Lecture 8/11/006 Thermodynamcs and statstcal mechancs n materals modellng II A bref résumé of the physcal concepts used n materals modellng Dr James Ellott.1
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationPaolo Giannozzi Scuola Normale Superiore, Piazza dei Cavalieri 7 I Pisa, Italy
Lecture Notes per l Corso d Struttura della Matera (Dottorato d Fsca, Unverstà d Psa, 2002): DENSITY FUNCTIONAL THEORY FOR ELECTRONIC STRUCTURE CALCULATIONS Paolo Gannozz Scuola Normale Superore, Pazza
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 7 Specal Relatvty (Chapter 7) What We Dd Last Tme Worked on relatvstc knematcs Essental tool for epermental physcs Basc technques are easy: Defne all 4 vectors Calculate c-o-m
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationMOLECULAR DYNAMICS ,..., What is it? 2 = i i
MOLECULAR DYNAMICS What s t? d d x t 2 m 2 = F ( x 1,..., x N ) =1,,N r ( x1 ( t),..., x ( t)) = v = ( x& 1 ( t ),..., x& ( t )) N N What are some uses of molecular smulatons and modelng? Conformatonal
More informationand Statistical Mechanics Material Properties
Statstcal Mechancs and Materal Propertes By Kuno TAKAHASHI Tokyo Insttute of Technology, Tokyo 15-855, JAPA Phone/Fax +81-3-5734-3915 takahak@de.ttech.ac.jp http://www.de.ttech.ac.jp/~kt-lab/ Only for
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationLecture 5.8 Flux Vector Splitting
Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More information1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:
68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationLaboratory 1c: Method of Least Squares
Lab 1c, Least Squares Laboratory 1c: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationSupplementary Notes for Chapter 9 Mixture Thermodynamics
Supplementary Notes for Chapter 9 Mxture Thermodynamcs Key ponts Nne major topcs of Chapter 9 are revewed below: 1. Notaton and operatonal equatons for mxtures 2. PVTN EOSs for mxtures 3. General effects
More information1. Mean-Field Theory. 2. Bjerrum length
1. Mean-Feld Theory Contnuum models lke the Posson-Nernst-Planck equatons are mean-feld approxmatons whch descrbe how dscrete ons are affected by the mean concentratons c and potental φ. Each on mgrates
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More informationReaxFF Potential Functions
ReaxFF Potental Functons Supportng nformaton for the manuscrpt A ReaxFF Reactve Force Feld for Molecular Dynamcs Smulatons of Hydrocarbon Oxdaton by Kmberly Chenoweth Adr CT van Dun and Wllam A Goddard
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationThermodynamics General
Thermodynamcs General Lecture 1 Lecture 1 s devoted to establshng buldng blocks for dscussng thermodynamcs. In addton, the equaton of state wll be establshed. I. Buldng blocks for thermodynamcs A. Dmensons,
More information8. Superfluid to Mott-insulator transition
8. Superflud to Mott-nsulator transton Overvew Optcal lattce potentals Soluton of the Schrödnger equaton for perodc potentals Band structure Bloch oscllaton of bosonc and fermonc atoms n optcal lattces
More informationEnergy, Entropy, and Availability Balances Phase Equilibria. Nonideal Thermodynamic Property Models. Selecting an Appropriate Model
Lecture 4. Thermodynamcs [Ch. 2] Energy, Entropy, and Avalablty Balances Phase Equlbra - Fugactes and actvty coeffcents -K-values Nondeal Thermodynamc Property Models - P-v-T equaton-of-state models -
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
More informationDynamics of a Superconducting Qubit Coupled to an LC Resonator
Dynamcs of a Superconductng Qubt Coupled to an LC Resonator Y Yang Abstract: We nvestgate the dynamcs of a current-based Josephson juncton quantum bt or qubt coupled to an LC resonator. The Hamltonan of
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More informationLaboratory 3: Method of Least Squares
Laboratory 3: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly they are correlated wth
More informationp 1 c 2 + p 2 c 2 + p 3 c p m c 2
Where to put a faclty? Gven locatons p 1,..., p m n R n of m houses, want to choose a locaton c n R n for the fre staton. Want c to be as close as possble to all the house. We know how to measure dstance
More informationarxiv:cond-mat/ v2 [cond-mat.mes-hall] 3 Jan 2006
arxv:cond-mat/0210519v2 [cond-mat.mes-hall] 3 Jan 2006 Non Equlbrum Green s Functons for Dummes: Introducton to the One Partcle NEGF equatons Magnus Paulsson Dept. of mcro- and nano-technology, NanoDTU,
More informationFermi Statistics and Fermi Surface. Sommerfeld Theory. 2.1 Fermi Statistics and Fermi Surface
erm Statstcs and erm Surface.1 erm Statstcs and erm Surface Snce Drude model, t too a quarter of a century for a breathrough to occur. That arose from the development of quantum mechancs and recognton
More informationarxiv: v2 [physics.chem-ph] 9 Dec 2014
Dervatve dscontnuty and exchange-correlaton potental of meta-ggas n densty-functonal theory F. G. Ech a) Department of Physcs, Unversty of Mssour-Columba, Columba, Mssour 65211, USA Mara Hellgren Internatonal
More informationRelaxation laws in classical and quantum long-range lattices
Relaxaton laws n classcal and quantum long-range lattces R. Bachelard Grupo de Óptca Insttuto de Físca de São Carlos USP Quantum Non-Equlbrum Phenomena Natal RN 13/06/2016 Lattce systems wth long-range
More informationLecture 4. Macrostates and Microstates (Ch. 2 )
Lecture 4. Macrostates and Mcrostates (Ch. ) The past three lectures: we have learned about thermal energy, how t s stored at the mcroscopc level, and how t can be transferred from one system to another.
More information