1. Optimization, Energy Landscapes, Protein Folding
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
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

2. Independent degrees of freedom
Schrödinger 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
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.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

3. Independent degrees of freedom
In 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
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

4. 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
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:
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

5. 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.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

6. 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
In this way, one commonly separates translational, rotational, and vibrational
degrees of freedom.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

7. The Born-Oppenheimer approximation
The Schrödinger equation for a molecule with n electrons, mass me, and N nuclei, masses Mt, is
(2.7)
kinetic energy of
nuclei
kinetic energy of
electrons
where x and X represent the electronic and nuclear coordinates, respectively,
and the potential energy is
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, 40 = 1,
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

8. Separation of degrees of freedom
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.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

9. Separation of degrees of freedom
They therefore considered an approximation for the total wavefunction:
where e(x;X) is a solution of the „electronic Hamiltonian“
(2.10)
: 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.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

10. Separation of degrees of freedom
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
(2.11)
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

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
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

12. 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
(2.12)
Hence we recover equations (2.10) and (2.11).
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

13. 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.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

14. 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.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

15. 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
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

16. General conical intersections and photochemistry
We may therefore write these two surfaces as
where we have used the fact that H is an Hermitian operator, so that
where the * denotes the complex conjugate.
The condition for the surfaces to intersect for some configuration X
is therefore that
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“.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

17. 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“.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

18. Rhodopsin: ultrafast isomerisation
Ben-Nun et al. PNAS 99, 1769 (2002)
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

19. Rhodopsin: ultrafast isomerisation
left topology yields
more productive decay
channel
Ben-Nun et al. PNAS 99, 1769 (2002)
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

20. 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.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

21. GFP: Equilibrium A  I  B
taken from Brejc et al. PNAS 94, 2306 (1997)
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding 1

22. Energielevels eines Atoms
Höchstes 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.
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding

23. Semiempirische QM: Konische Durchschneidungen
Energie im elektronisch
angeregten Zustand
Energie im elektronischen
Grundzustand.
Konische Durchschneidung:
In bestimmten Konformationen
können die Energien für zwei
elektronische Zustände gleich
(bzw. fast gleich) sein
Das Molekül kann ohne
Energieabgabe (Photon) direkt
in den anderen Energiezustand
übergehen.
Hier: Für die rosa Konformationen
sind die Energien des Grund-
zustands und des angeregten
S1-Zustands gleich
Wenn diese Konformationen
energetisch zugänglich sind,
erscheinen diese Zustände
dunkel, fluoreszieren also
nicht.
Weber, Helms et al. PNAS 96, 6177 (1999)
Frage: welche Punkte sind bei Raumtemperatur thermisch erreichbar?
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding 1

24. GFP: Photophysikalisches Termschema
Neutrales Inter- Negatives Zwitterionisches
Chromophor mediat Chromophor Chromophor
(dunkel)
Weber, Helms et al. PNAS 96, 6177 (1999)
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding 1

25. Optimization, Energy Landscapes, Protein Folding
GFP more accurate
chromophore
is pyramidically
deformed at
conical intersection
Toniolo et al. Faraday Discuss. 127, 149 (2004)
10. Lecture SS 20005
Optimization, Energy Landscapes, Protein Folding 1