Atom and molecular structure Quantum level of matter

Size: px

Start display at page:

Download "Atom and molecular structure Quantum level of matter"

Melvin Rose
5 years ago
Views:

1 Computer imulation of advanced material International Summer School Atom and molecular tructure Quantum level of matter Alexander Nemukhin Department of Chemitry, M.V. omonoov Mocow State Univerity and N.M. Emanuel Intitute of Biochemical Phyic, Ruian Academy of Science

2 Computer imulation of advanced material International Summer School Atom and molecular tructure Quantum level of matter Textbook material and illutration Chemical tructure theory i valid (molecule, chemical bond, ) Model ytem compoed of large number of atom are of interet Protein may be conidered a advanced material (at leat for biotechnology)

3 θ 0 r 0 M N σ σ ε φ + + φ + + θ θ + MN 6 MN MN 1 MN MN MN MN MN N M torion l l l, l l,1 j0 j bend j j i0 tretche i i i MM r r 4 r q q... ) co (1 V ) co (1 V ) ( k ) r (r k E -,,,q,v,v,,k,r k MN MN M l, l,1 j0 j i0 i σ ε θ parameter Мolecular mechanical (MM) model The ytem i compoed of atom

4 R w r i Ψ Ψ < < iw i w w v w wv v w j i ij w R w i r QM r R Z R Z Z r 1 M 1 1 Ĥ E Ĥ w i Electronic tructure model H ES C C Numerical olution of matrix equation Molecule are table ytem of nuclei and electron A uch, their tructure and dynamic hould be governed by Quantum Mechanic

5 Computer imulation of advanced material Why do we need quantum mechanic?

protein Ser05 Glu S1 S1* anion S1* S1 Hi148 Arg96 S0 S0 S0

6 Why do we need quantum mechanic? - Excited electronic tate are involved Example: fluorecent v protein Ser05 Glu S1 S1* anion S1* S1 Hi148 Arg96 S0 S0 S0 anion S0 neutral Bravaya K.B., Grigorenko B.., Nemukhin A.V., Krylov A.I., Acc Chem Re, 45, 65 (01)

7 Why do we need quantum mechanic? - Cleavage and formation of chemical bond are involved Example: hydrolyi of adenoine triphophate (ATP) in motor protein Arg38 Ser36 Ser37 Wat Thr186 Mg + Wat1 Wat P γ O βγ ATP Glu459 Wat Ser181 Grigorenko B.., Nemukhin A.V., et al., Proc Natl Acad Sci USA, 104, 7057 (007)

8 Why do we need quantum mechanic? - Force field parameter may be not accurate enough GTP Grigorenko B.., Shadrina M.S., Nemukhin A.V., et al., Biochim Biophy Acta, 1784, 1908 (008)

9 Computer imulation of advanced material International Summer School Atom and molecular tructure Quantum level of matter Baic of quantum mechanic

10 Baic of quantum mechanic General conideration Quantum and claical mechanic Wavefunction Obervable Hydrogen atom Spin

11 Quantum mechanic i the theory of the behavior of microcopic object, including electron and nuclei, for which the action S t 1 0 t ( q 1,..., qn, t) dt i comparable to the Planck contant ћ J

12 Claical Mechanic The agrangian function ( T-V ) The Euler-agrange equation The Hamiltonian ( HT+V ) d dt H q i q i 0 (i1,,n) q1,..., qn, p,..., pn, t) ( 1 The Hamilton' equation p i H q i q i H p i The Hamilton Jacobi formulation S S S H ( q1,..., qn,,...,, t) + q q t 1 n 0

13 Claical mechanic S» ћ Correpondence principle Quantum mechanic S ~ ћ Sytem can be characteried by a function of coordinate and time S( q 1,..., qn, t) Ψ( q 1,..., q, t) S define tate of the ytem Sytem can be characteried by function of coordinate and time n Each Ψ refer to a tate of the ytem Thi function S can be found a a olution of the differential equation S S S H ( q1,..., qn,,...,, t) + q q t 1 Knowing function S one can find trajectorie and to compute obervable n 0 Function Ψ can be found a olution of the differential equation i Ψ t Hˆ Ψ Knowing function Ψ one can compute obervable

14 Claical and Quantum Mechanic Keyword: Sytem State of the ytem Wavefunction of a tate Obervable for a particular tate

15 Quantum Mechanic: Keyword Sytem Sytem the Hamiltonian operator Ĥ A only we elect a ytem for an analyi we can write down an explicit expreion for the Hamiltonian operator. Example: A particle of ma m in a one- ˆ dimentional potential V(x) H + V ( x) m x One-elctron atom: an electron in the field of a nucleu Hˆ µ x + y + Ze r

16 Quantum Mechanic: Keyword State of the ytem An eential feature of QM i that certain parameter of the ytem can take on dicrete value varying from one tate to another by quantum jump. Example: Few tate of the hydrogen atom State Total energy (10 0 J/atom) Electronic angular momentum p ħ

17 Thee jump are oberved in experimental pectrocopy State and p of the hydrogen atom Total energy (10 0 J/atom) State 1 of the hydrogen atom

18 Claical and Quantum Mechanic Keyword: Sytem State of the ytem Wavefunction of a tate Obervable for a particular tate

19 Quantum Mechanic: Keyword Wavefunction The tate of a ytem i decribed by a wavefunction of the coordinate and the time Ψ(q 1,, q n,t) The probability: Ψ *( r, t) Ψ( r, t) dxdyd Ψ mut atify mathematical condition: Single-value Continuou Quadratically integrable The probability that the particle i in the volume element dxdyd located at, at time t. r Ψ *( r, t) Ψ( r, t) dxdyd 1

20 Quantum Mechanic: Keyword Wavefunction Why probabilitie?

21 The Heienberg uncertainty principle: The more preciely the poition i determined, the le preciely the momentum i known in thi intant, and vice vera. large pread in p mall pread in x. probability of p p p mall pread in p large pread in x. probability of x x x probability of p p p Quantum particle do not travel along trajectorie. probability of x x x

22 Keyword Wavefunction : Superpoition principle The negative content (andau, ifhchit) of the uncertainty principle i balanced by the poitive content of the uperpoition principle: If Ψ 1, Ψ,, Ψ n are the poible tate of a ytem, then the linear combination of thee tate i alo a poible tate of the ytem Phyical conequence: The equation for Ψ mut be linear, e.g., Ψ( x, t) i t m Ψ( x, t) + V ( x, t) Ψ( x, t) x Either the Schrödinger equation, or the uperpoition principle hould be potulated

23 Potulate of Quantum Mechanic 1. Probabilitie (due to the uncertainty principle): Ψ *( r, t) Ψ( r, t) dxdyd The probability that the particle. The uperpoition principle: i in the volume element dxdyd located at, at time t. If Ψ 1, Ψ,, Ψ n are the poible tate of a ytem, then the linear combination of thee tate i alo a poible tate of the ytem. and 3. For every obervable mechanical quantity of a ytem, there i a correponding linear Hermitian operator aociated with it. r

24 Quantum Mechanic Keyword: Sytem State of the ytem Wavefunction of a tate Obervable for a particular tate

25 Keyword Obervable For every obervable mechanical quantity of a ytem, there i a correponding linear Hermitian operator aociated with it. To pecify thi operator, write down the claical expreion for the obervable in term of Carteian coordinate and the correponding linear momentum, and then replace each coordinate x by the operator x and each momentum component p x by the operator p x i x

26 Keyword Obervable Some mechanical quantitie and their operator Poition (x) x inear momentum (p x ) p x i x Angular momentum ( xp y -yp x ) i( x y y ) x Total energy (T+V) ˆ H + V ( x, m + + x y y,, t)

27 Keyword Obervable If a ytem i in a tate decribed by a normalied wavefunction Ψ, then the average value of the obervable A in thi tate i given by A Ψ Ψ AΨ The Hermitian operator enure a real number for. We talk on average value by the ame reaon a on probabilitie (the uncertainty principle). A

28 Keyword Obervable : Eigenfunction and eigenvalue Finding eigenfunction and eigenvalue of operator aociated with obervable i one of the major goal of QM. Hˆ T electron + T nuclei + V V e e + nn + V HˆΨ Solution of the time-independent Schrödinger equation allow one to find the total energie E i of the ytem under tudy. en i E i In particular, for any molecule compoed of electron (e) and nuclei (n) Ψ i The differential equation mut be augmented by the boundary condition for the wavefunction (ingle-value, continuou, ) ince they decribe wave of probabilitie.

29 Keyword Obervable : Orbital angular momentum Eigenfunction and eigenvalue of the component i( x y ) i y x ϕ x θ ϕ r y Differential equation: Φ ( ϕ) l Φ ( ϕ) m Condition: Φ( ϕ + π ) Φ( ϕ) Φ Φ Anwer: π 0 Φ ( ϕ) Φ( ϕ) dϕ 1 l m 1 Φm( ϕ) exp( imϕ) π m 0, ± 1, ±,... * m +3ћ +ћ +ћ 0 -ћ -ћ -3ћ quantum jump

30 Keyword Obervable : Meaurement Important quetion of the QM theory: What do we know about A in the tate characteried by the wavefunction Ψ? Anwer: If Ψ happen to coincide with one of the eigenfunction of A then in thi particular tate we know A preciely, and the only reult of the meaurement i the correponding eigenvalue of A If Ψ doe not coincide with any of the eigenfunction of A then we can predict only the averaged value of A: A Ψ Ψ AΨ A probability of meauring a particular value a k in the tate Ψ i w k k Φ Ψ where Φ k i the correponding eigenfunction of A.

31 Keyword Obervable : Back to the uncertainty principle Another important quetion of the QM theory: Can we know two mechanical variable A and B imultaneouly and preciely? Anwer: Two obervable A and B are principally imultaneouly meaurable (have the common et of eigentate) if and only if their correponding operator commute. More pecifically: The commuting operator have the ame et of eigenfunction, and if the wavefunction of a particular tate coincide with one of thee eigenfunction, one may know A and B imultaneouly and preciely in thi tate. AB BA When the operator do not commute, one can never know A and B imultaneouly and preciely in any tate.

32 Keyword Obervable : Back to the uncertainty principle Quantitative ide of the uncertainty principle The uncertaintie ΔA and ΔB of any two obervable in any phyical tate Ψ atify the inequality 1 A B Ψ ( AB BA) Ψ Poition and momentum: hence, x p x probability of p xpx pxx i large pread in p mall pread in x. p p mall pread in p large pread in x. probability of x x x probability of p p p probability of x x x

33 Keyword Obervable : Back to the uncertainty principle Angular momentum operator ) ( ) ( ) ( y x x y x y y x x y y x i yp xp x x i xp p y y i p yp + + The commutator i i i x y y y y y x x x x x y y x

34 Keyword Obervable : Back to the uncertainty principle All three component of angular momentum cannot be meaured imultaneouly, which tell u that there doe not exit any phyical tate in which the direction of angular momentum i definite. However, there do exit tate in which the magnitude of angular momentum i definite along with one component. y x One can know imultaneouly and preciely and while no knowledge on x and y i allowed. The phyical reaon are traced back to the uncertainty principle.

35 Keyword Obervable : Example of commuting operator The commuting operator and common et of eigenfunction. i ϕ 1 1 inθ + inθ θ θ in have a θ ϕ The pherical harmonic The olution Ylm ( θ, ϕ) ( m) Ylm ( θ, ϕ) Y ( θ, ϕ) l( l + 1) Y ( θ, ϕ) lm m 0, ± 1, ±,..., ± l l 0,1,,... lm

36 Hydrogen atom The one-electron atom: The ytem - an electron with the charge e and ma μ in the field The Hamiltonian operator θ x y ϕ r r Ze r V ) ( r Ze r r r r r H + µ µ State of the ytem are decribed by the olution of the equation ),, ( ),, ( ϕ θ ϕ θ r E r H Ψ Ψ Since H and, are the pairwie commuting operator there exit tate in which the total energy, the magnitude of angular momentum, and the component of angular momentum are definite. 0 0, 0, H H H H

37 Hydrogen atom H H 0, H H 0, 0 Therefore, the energy eigentate may alo be choen to be eigentate of the angular momentum operator Ψ E ( r, θ, ϕ) R ( r) Y ( θ, ϕ) El lm Y Y lm lm ( θ, ϕ) ( θ, ϕ) ( m) Y l( l m 0, ± 1, ±,..., ± l l 0,1,,... lm + 1) Y ( θ, ϕ) lm ( θ, ϕ)

38 Hydrogen atom Therefore, the energy eigentate may alo be choen to be eigentate of the angular momentum operator Ψ E ( r, θ, ϕ) R ( r) Y ( θ, ϕ) El lm Radial eigenfunction R El (r) and energie E are computed from the ordinary differential equation d dr l( l + 1) r + µr dr dr µ r Boundary condition: R(r) and dr dr Ze r be quadratically integrable R ( r) r dr <. The anwer: R ( r) Polnl( r)exp( β r) 4 µ E n nl 0 E R( r) 0 mut be continuou, R(r) mut Z e n1,, l0,1,,n-1 m-l, -l+1, l n n

39 Hydrogen atom: State Quantum Mechanic: Keyword State of the ytem An eential feature of QM i that certain parameter of the ytem can take on dicrete value varying from one tate to another by quantum jump. Example: Few tate of the hydrogen atom State 1 p Total energy (10 0 J/atom) Electronic angular momentum 0 0 ħ Z e 4 µ E n n n1,, l0,1,,n-1 m-l, -l+1, l 1 n1, l0, m0 n, l0, m0 p n, l1, m-1,0,1 3 n3,l0,m0

40 Hydrogen atom: Image of wavefunction 1 n1, l0, m0 R nl R nl r R nl

41 Hydrogen atom: Image of wavefunction n, l0, m0 R nl R nl r R nl

42 Hydrogen atom: Image of wavefunction p n, l1, m0 R nl R nl r R nl

43 Hydrogen atom: Image of wavefunction 3 n3, l0, m0 R nl R nl r R nl

44 Hydrogen atom: Image of wavefunction 3p n3, l1, m0 R nl R nl r R nl

45 Hydrogen atom: Image of wavefunction 3d n3, l, m0 R nl R nl r R nl

46 Hydrogen atom: Atomic Orbital (AO) 1 p 3 3p 3d

47 Spin In the non-relativitic quantum mechanic, we have to aign to every elementary particle an angular momentum which i not related to poible orbiting of the particle. An internal angular momentum, the vector with component x, y,, i called pin. Unlike orbital angular momentum thee component cannot be expreed in term of Carteian coordinate and linear momenta. The quare of the pin length i a characteritic feature of a particle: for every electron in all tate it ha the value 1/ 3 ( + 1) 4

48 Spin i i i x y y y y y x x x x x y y x The theory of pin i contructed by analogy with the theory of orbital angular momentum. Important: the commutator relation only two quantitie are meaurable and have a common et of eigenvector and + m m m m m,,, 1) (, For electron: ½ (alway) m -½, +½, m

49 Spin i i i x y y y y y x x x x x y y x The theory of pin i contructed by analogy with the theory of orbital angular momentum. Important: the commutator relation only two quantitie are meaurable and have a common et of eigenvector and + m m m m m,,, 1) (, For electron: ½ (alway) m -½, +½ Orbital angular momentum, m 0,1,,...,..., 1, 0, ), ( 1) ( ), ( ), ( ) ( ), ( ± ± ± + l l m Y l l Y m Y Y lm lm lm lm ϕ θ ϕ θ ϕ θ ϕ θ

50 Spin Two poible pin tate of an electron +ћ/ 3ћ/ 3ћ/ -ћ/ 1/, m 1/ α β 1/, m 1/

51 Spin Application of addition of angular momentum -electron pin vector (j 1 ½, j ½) The total pin of two electron may be either ½-½ 0 (inglet tate), or ½+½1 (triplet tate) Two poible pin tate of an electron +ћ/ 3ћ/ 3ћ/ -ћ/ α 1/, m 1/ β 1/, m 1/ Singlet tate -electron pin vector 1 0,0 { α ( 1) β () β (1) α ()} Triplet tate -electron pin vector 1,1 α (1) α (1) () 1 1,0 { α (1) β 1, 1 β β () () + β (1) α () }

52 Interchange Hypothei For Identical Particle Interchanging the poition of two identical particle doe not change the phyical tate. Identical (elementary) particle have the ame parameter the ma, charge, and magnitude of pin momentum. Poition r and pin projection σ are conidered a coordinate. The wavefunction mut be either ymmetric or antiymmetric with repect to an interchange of poition and pin projection of any pair of identical particle: Ψ( rσ1,..., riσ i,... rjσ j,..., rnσ N ) ±Ψ( r1 σ1,..., rjσ j,... riσ i,..., rnσ 1 N )

53 The Spin Statitic Theorem Sytem of identical particle with integer pin 0; 1; ; are decribed by wavefunction that are ymmetric under the interchange of particle coordinate and pin. Sytem of identical particle with half-integer pin 1/; 3/; are decribed by wavefunction that are antiymmetric under the interchange of particle coordinate and pin. Particle with integer pin are known a Boon. Particle with halfinteger pin are known a Fermion. Example include, in particular, electron, and proton. Therefore, electronic wavefunction mut atify a condition Ψ( rσ1,..., riσ i,... rjσ j,..., rnσ N ) Ψ( r1 σ1,..., rjσ j,... riσ i,..., rnσ 1 N )

54 Quantum Mechanic for the Electronic Structure Theory Sytem State of the ytem Correpondence principle Claical mechanic Wavefunction Superpoition principle Hilbert pace Obervable Uncertainty principle Hydrogen atom Orbital Spin Spin coupling Interchange hypothei

55 R w r i Ψ Ψ < < iw i w w v w wv v w j i ij w R w i r QM r R Z R Z Z r 1 M 1 1 Ĥ E Ĥ w i Molecule are table ytem of nuclei and electron

56 R w r i Ψ Ψ < < iw i w w v w wv v w j i ij w R w i r QM r R Z R Z Z r 1 M 1 1 Ĥ E Ĥ w i Molecule are table ytem of nuclei and electron The Born-Oppenheimer approximation Moving further To a high degree of accuracy we can eparate electron and nuclear motion due to larger mae of nuclei

57 The Born-Oppenheimer Approximation Hˆ H ˆel T electron electron + T + T nuclei to the nuclear equation T nuclei + V + V V e e + nn + V e e + nn + V en V en Electronic equation ˆ H el ψ el (r;r) E el ψ el (r;r) BO approximation lead to the idea of a potential energy urface U(R) E el + V NN n-n

58 Solution of the nuclear quantum equation allow u to determine a large variety of molecular propertie. An example are vibrational pectra.

59 Example: calculated vibration of the -CO group in the bacteriochlorophyll of photoynthetic reaction center 59

60 Example: calculated vibration with the imaginary frequency on the route of retinal iomeriation in rhodopin С 11 С 1 Rhodopin Bathorhodopin Khrenova M.G., Bochenkova A.V., Nemukhin A.V., Protein, 78, 614 (010)

61 About electronic equation etc The lecture by Profeor Gian Paolo Brivio Ab-initio & DFT on July 18

62 Concluding example How doe it work?

63 Concluding example How doe it work? Chromophore Green Fluorecent Protein НВМО UMO гидроксибензилиден-имидазолинон hydroxybenylidene-imidaoline ВЗМО HOMO

70 Concluding example How doe it work for large ytem? Practically ueful tool - Quantum mechanical molecular mechanical (QM/MM) approach QM MM Primary interet Coupling

$Experiment (PDB ID - 1GF): X-ray diffraction (PH 7,$

71 The reult of QM/MM imulation: Geometry of the GFP active ite Theory: QM/MM (DFT(PBE0/6-31G*)/Amber) Experiment (PDB ID - 1GF): X-ray diffraction (PH 7, reolution 1.9A) neutral anion

76 Computer imulation of advanced material International Summer School Atom and molecular tructure Quantum level of matter Textbook material and illutration Software QM: GAMESS(US), Firefly, NWChem, CPK QM/MM: GAMESS(US)-Tinker

77 Concluding Remark Thi i a part of the lecture coure Quantum Mechanic and Molecular Structure for tudent of the Chemitry Department of the M.V. omonoov Mocow State Univerity Part of the tutorial lecture at the workhop Mathematical and Computational Approache to Quantum Chemitry (Intitute of mathematic and it application, Minneapoli, 008) Thank to co-author of the paper

p. (The electron is a point particle with radius r = 0.)

p. (The electron is a point particle with radius r = 0.) - pin ½ Recall that in the H-atom olution, we howed that the fact that the wavefunction Ψ(r) i ingle-valued require that the angular momentum quantum nbr be integer: l = 0,,.. However, operator algebra