10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding1 V10: beyond the Born-Oppenheimer approximation – coupled energy surfaces Todays lecture

10. Lecture SS 20005

Optimization, Energy Landscapes, Protein Folding 1

V10: beyond the Born-Oppenheimer approximation –coupled energy surfaces

Today‘s lecture is designed after Chapter 2 of the Wales book

- Born-Oppenheimer (BO) approximation potential energy landscape

BO greatly simplifies the construction of partition functions

- neglect of terms that couple together electronic and nuclear degrees of

freedom

separate Schrödinger equation into independent nuclear and electronic parts

nuclear motion is governed entirely by a single PES for each electronic state

investigate situations in photochemistry where BO breaks down



Independent degrees of freedomSchrödinger equation (SE)

The „Hamiltonian“ H is the operator of the total energy H = T + V, where

T is the kinetic energy,

V is the potential energy.

is the electronic wave function,

E are the energy eigenvalues.

The wave function if typically expressed as linear combination of atomic orbitals n

Eˆ

nnc

The optimal coefficients are obtained by the „variational principle“:

given a normalized wave function |> that satisfies the appropriate boundary

condition (usually the requirement that the wave function vanishes at infinity),

the the expectation value of the Hamiltonian is an upper bound to the exact

ground state energy:

Therefore one just needs to optimize the coefficients cn to minimize this integral.

0ˆ1 EH



Independent degrees of freedomIn quantum mechanics, the momentum is expressed as

This means that coordinate and momentum „do not commute“ (vertauschen nicht)

If more than one coordinate is involved, the SE is a partial differential equation

Most common method of solution for PDEs: try separating the variables.

E.g. suppose that the Hamiltonian can be separated into two parts, the first involving

only coordinate x, the second involving only coordinate y, then

yxyHxHyxEyxyxH yx ,ˆˆ,,,ˆ

xp

ixxx

xx

xpx

,,



Independent degrees of freedom

This separation allows us to find a solution with

Since has no effect on Y(y), and has no effect on X(x),

we obtain

yYxXyx ,

xH xˆ yH y

ˆ

const

ˆˆ

thatsoˆˆ

EyY

yYyH

xX

xXxH

yYxEXyYyHxXxXxHyY

yx

yx

This equation must hold for any values of x and y.

Because both terms on the left are independent of eachother, they must

both be equal to constants Ex and Ey:

EE

yYEyYyH

xXExXxH

y

yy

xx

xE

withˆ

ˆ



Separation of degrees of freedom

for independent degrees of freedom, where the Hamiltonian contains no

terms that couple the different coordinates together,

the total wavefunction and total energy can be written as a product and sum,

respectively, using the wavefunctions and energies obtained for the separate

degrees of freedom.

General case: the coupling is never exactly equal to zero,

but can be close to zero.



Partition functions for separable degrees of freedom

For a system with fixed temperature T, volume V, and number of particles, N,

the partition function is

with the Boltzmann constant k, and the sum is over all possible states of the

system.

Assuming two separable degrees of freedom, each energy level can be written

as Ei = Ex + Ey

and

one can decompose

i

kTEi

eTVNZ ,,

kTE

kTE

kTE yxi

eee

TVNZTVNZTVNZ yx ,,,,,,

In this way, one commonly separates translational, rotational, and vibrational

degrees of freedom.



The Born-Oppenheimer approximationThe Schrödinger equation for a molecule with n electrons, mass me, and N nuclei,

masses Mt, is XxXxXx ,,,221 1

22

22

total

N

t

N

ti

et

t

EVmM

kinetic energy of nuclei

kinetic energy of electrons

where x and X represent the electronic and nuclear coordinates, respectively,

and the potential energy is

tiall ji st ts

st

ijit

t

r

ZZ

rr

ZeV

,0

2 1

4Xx,

where Zt : the nuclear charge (atomic number) of nucleus t

e : the charge on a proton.

rij , rit and rts are the distances between two electrons, an electron and a nucleus,

or between two nuclei.

V(x,X) is essentially the Coulomb interaction between electrons and nuclei.

It is convenient to switch to atomic units where e = 1, me = 1, 40 = 1,

(2.7)

1




This equation cannot straightforwardly solved by separating the variables due to

the distance terms between electrons and nuclei in the potential energy.

Because

Max Born and Robert Oppenheimer reasoned that the electron density should

adjust almost instantaneously to changes in the positions of the nuclei.

From a classical viewpoint, the electrons are expected to move much faster than

the nuclei.

electronproton mm 1836




They therefore considered an approximation for the total wavefunction:

XXx;Xx, ne where e(x;X) is a solution of the „electronic Hamiltonian“

Xx;XXx; eeen VTH ]ˆˆ[

: total Hamiltonian operator in eq.(2.7),

: nuclear kinetic energy operator (first term in 2.7)

: is a function of the electronic coordinates x

(actually it only depends upon the nuclear positions X parametrically,

because 2.10 is solved for a particular nuclear geometry.

write to show that different electronic wavefunctions

and energies are obtained for different nuclear configurations.

The nuclear coordinates X only appear in Ve(X) and the wavefunction

e(x;X) in the form of fixed points.

(2.10)

XXx; ee Vand

Xx;e

nT

H

ˆ

ˆ




The potential energy surface defines the variation of the electronic energy

Ve(X) with the nuclear geometry.

Often, the „e“ is omitted, and we simply refer to a potential energy surface V(X).

This implicitly assumes that we refer to the PES of the electronic ground state.

Remember that there exist different solutions of (2.10) that represent excited

electronic states.

If Ve(X) defines an effective potential for the nuclei, then the appropriate

Schrödinger equation for the nuclear wavefunction, n(X), is

XXXX ntotalnen EVT ]ˆ[ (2.11)



Alternative derivation of the BO approximation

Alternatively, we can derive the electronic and nuclear BO equations (2.10) and

(2.11) by separating the variables if certain terms are neglected.

Substituting into (2.7) gives XXx;Xx, ne

XXx;

XXx;Xx,Xx;X

Xx;XXx;X

XXx;

XXx;Xx

netotal

nee

n

iin

N

tetntetn

t

n

N

tte

t

ne

N

t

N

tit

t

E

V

M

M

VM

1

2

1

2

1

2

1 1

22

2

1

22

1

2

1

,2

1

2

1



Alternative derivation of the BO approximation

Neglecting all the terms involving derivatives of

with respect to nuclear coordinates,

i.e. and

and dividing by

gives

Xx;e

etn 2

etnt

XXx; ne

N

tetotal

nt

ntn

i e

ei VEM

V1

2

1

2

2,

2X

X

XXx

Xx;

Xx;

Hence we recover equations (2.10) and (2.11).

(2.12)



Breakdown of the BO approximation

PES only exist within the Born-Oppenheimer approximation.

If the approximation were exact, then H – D would have no dipole moment,

because the extra neutron in the frozen deuterium nucleus would not affect the

electrons.

In fact, H – D has a very small dipole moment of 10-4 D (a water molecule has a

dipole of 1.85 D) the BO approximation works very well for H – D.

However, the neglected terms in (2.12) are only small if the electronic

wavefunction is a slowly varying function of the nuclear coordinates.

This approximation may break down if the electronic wavefunction is

degenerate, or nearly degenerate, because the neglected terms may cause

a significant interaction between the BO surfaces.

Coupling may occur due to the Renner and Jahn-Teller effects.



Adiabatic approximation

The separation of nuclear and electronic motion is sometimes called an

adiabatic approximation:

the nuclear dynamics are assumed to be slow enough so that separate

electronic states can be defined where the nuclei move according to a single

adiabatic PES generated by the electrons.

Processes in which a system moves between different adiabatic PES

corresponding to different electronic states, are therefore termed nonadiabatic.

Breakdown of the BO approximation can result in nonadiabatic transitions

without the absorption or emission of radiation.

Adiabatic surface crossings via conical intersections or avoided crossings are of

central importance in photochemistry.



General conical intersections and photochemistry

Until recently, surface crossings not arising from symmetry requirements have

been relatively neglected due to a „non-crossing“ rule which actually only

applies to diatomic molecules.

To derive this rule, Edward Teller considered two electronic states with

wavefunctions A and B which are functions of the nuclear coordinates X and

are orthogonal to all the other electronic states, and to each other.

For any given X the two corresponding PES are determined by the two

eigenvalues of the matrix

where the matrix elements are

XX

XX

BBBA

ABAA

HH

HH

rr

ddH

HH

BA

BAAB

ˆ

ˆ

*

xx EA



General conical intersections and photochemistry

We may therefore write these two surfaces as

22 42

1

2XXX

XXX ABBBAA

BBAA HHHHH

E

where we have used the fact that H is an Hermitian operator, so that

*XX BAAB HH

where the * denotes the complex conjugate.

The condition for the surfaces to intersect for some configuration X

is therefore that XX BBAA HH and H(X) = 0.

For a diatomic molecule, there is only one degree of freedom, the distance,

so that the two conditions could only be satisfied „accidentally“.



Conical intersections

For a polyatomic molecule, there are more degrees of freedom,

and crossings of different electronic state surfaces may occur.

When two surfaces intersect, this is termed „conical intersection“.

Examples are the ultrafast twisting of retinal and of the GFP chromophore.

If they only get close, this is termed „nonadiabatic crossing“.



Rhodopsin: ultrafast isomerisation

Ben-Nun et al. PNAS 99, 1769 (2002)



Rhodopsin: ultrafast isomerisation

Ben-Nun et al. PNAS 99, 1769 (2002)

left topology yieldsmore productive decaychannel



Appetizer: das grün fluoreszierende Protein

Die Alge Aequorea victoria enthält ein Protein, das

sogenannte grün fluoreszierende Protein, das für ihre

grüne Fluoreszenz verantwortlich ist.

Dieses Protein absorbiert das von einem anderen Protein,

XYZ emittierte blaue Licht, und emittiert grünes Licht.

Dreidimensionale Struktur von GFP.

Für die Fluoreszenz verantwortlich ist das kleine

aromatische Ringsystem in seiner Mitte.



taken from Brejc et al. PNAS 94, 2306 (1997)

GFP: Equilibrium A I B



Energielevels eines AtomsHöchstes unbesetztes Molekülorbital Niedrigstes unbesetztes Molekülorbital

Helms, Winstead, Langhoff, J. Mol. Struct. (THEOCHEM) 506, 179 (2000)

Bei Lichtanregung (Absorption eines Photons)

wird ein Elektron aus dem HOMO in das

LUMO angeregt (vereinfachte Darstellung,

HOMO LUMO Übergang macht 90% der

Anregung aus).

Später wird ein Photon emittiert. Seine

Wellenlänge (Energie) entspricht der Energie-

differenz von angeregtem Zustand und

Grundzustand.



Weber, Helms et al. PNAS 96, 6177 (1999)

Semiempirische QM: Konische Durchschneidungen

Energie im elektronisch angeregten Zustand

Energie im elektronischenGrundzustand.

Konische Durchschneidung:In bestimmten Konformationenkönnen die Energien für zweielektronische Zustände gleich(bzw. fast gleich) seinDas Molekül kann ohneEnergieabgabe (Photon) direktin den anderen Energiezustandübergehen.

Hier: Für die rosa Konformationen sind die Energien des Grund-zustands und des angeregten S1-Zustands gleich Wenn diese Konformationenenergetisch zugänglich sind,erscheinen diese Zuständedunkel, fluoreszieren alsonicht.Frage: welche Punkte sind bei Raumtemperatur thermisch erreichbar?



Weber, Helms et al. PNAS 96, 6177 (1999)

GFP: Photophysikalisches Termschema

Neutrales Inter- Negatives ZwitterionischesChromophor mediat Chromophor Chromophor

(dunkel)



Toniolo et al. Faraday Discuss. 127, 149 (2004)

GFP more accurate

chromophoreis pyramidicallydeformed atconical intersection

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding1 V10: beyond the Born-Oppenheimer approximation – coupled energy surfaces Todays lecture