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

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding2 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 c n to minimize this integral.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding3 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

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding4 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 E x and E y :

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding5 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.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding6 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 E i = E x + E y and one can decompose In this way, one commonly separates translational, rotational, and vibrational degrees of freedom.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding7 The Born-Oppenheimer approximation The Schrödinger equation for a molecule with n electrons, mass m e, and N nuclei, masses M t, is 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 Z t : the nuclear charge (atomic number) of nucleus t e : the charge on a proton. r ij, r it and r ts 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, m e = 1, 4 0 = 1, (2.7)

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding8 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.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding9 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 : 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 V e (X) and the wavefunction e (x;X) in the form of fixed points. (2.10)

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding10 Separation of degrees of freedom The potential energy surface defines the variation of the electronic energy V e (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 V e (X) defines an effective potential for the nuclei, then the appropriate Schrödinger equation for the nuclear wavefunction, n (X), is (2.11)

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding11 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

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding12 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 Hence we recover equations (2.10) and (2.11). (2.12)

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding13 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 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.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding14 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.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding15 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

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding16 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.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding17 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.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding18 Rhodopsin: ultrafast isomerisation Ben-Nun et al. PNAS 99, 1769 (2002)

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding19 Rhodopsin: ultrafast isomerisation Ben-Nun et al. PNAS 99, 1769 (2002) left topology yields more productive decay channel

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding20 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.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding21 taken from Brejc et al. PNAS 94, 2306 (1997) GFP: Equilibrium A I B

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding22 Energielevels eines Atoms Höchstes unbesetztes MolekülorbitalNiedrigstes 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.

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding23 Weber, Helms et al. PNAS 96, 6177 (1999) 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. Frage: welche Punkte sind bei Raumtemperatur thermisch erreichbar?

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding24 Weber, Helms et al. PNAS 96, 6177 (1999) GFP: Photophysikalisches Termschema NeutralesInter-Negatives Zwitterionisches ChromophormediatChromophorChromophor (dunkel)

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10. Lecture SS 20005Optimization, Energy Landscapes, Protein Folding25 Toniolo et al. Faraday Discuss. 127, 149 (2004) GFP more accurate chromophore is pyramidically deformed at conical intersection

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