Computational Chemistry. Ab initio methods seek to solve the Schrödinger equation.

Transcription

1 Theory Computational Chemistry Ab initio methods seek to solve the Schrödinger equation. Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory (DFT) attempts to solve for the electron density function. Semi-empirical methods use approximate Hamiltonians that are partly parameterized using experimental data. Molecular mechanics uses Newtonian mechanics and empirically parameterized force fields. Jay Taylor (ASU) APM Lecture 3 Fall / 56

2 Theory The Time-Dependent Schrödinger Equation The time-dependent Schrödinger equation can be written as where Ψ(x, t) i = HΨ(x, t) t = J s is the reduced Planck s constant; the operator H is the system Hamiltonian; Ψ(, t) is the wavefunction of the system at time t. The modulus of the wavefunction, Ψ(, t) 2, is often interpreted as the probability density of the system at time t. Jay Taylor (ASU) APM Lecture 3 Fall / 56

3 Theory The Time-Independent Schrödinger Equation If H is time-indepedent, then we can seek an eigenfunction expansion Ψ(x, t) = c n Ψ n (x)e i En t n=1 where Ψ n satisfies the eigenvalue problem HΨ n = E n Ψ n and E n is interpreted as the energy of the stationary configuration Ψ n. The eigenfunction Ψ 0 corresponding to the smallest eigenvalue E 0 is called the ground state of the system. Jay Taylor (ASU) APM Lecture 3 Fall / 56

4 Theory The Hamiltonian of a Molecule The Hamiltonian H can be written as the sum of the kinetic and potential energy operators H = T + V, where for a (non-relativistic) molecule T = 2 2 V = 1 4πɛ 0 i i<j 1 m i e i e j r ij ( 2 x 2 i + 2 yi zi 2 ) (electrostatic potential). Here i, j range over nuclei and electrons, m i and e i are the mass and charge of the i th particle, and ɛ 0 is the vacuum permittivity. Jay Taylor (ASU) APM Lecture 3 Fall / 56

5 Hydrogen-like atom Hydrogen-like Atom In a few cases, Schrödinger s equation can be solved analytically. For a single nucleus with mass M and charge +Ze with a single electron (of mass m), we have: ( 2 ) 2(M + m) 2 CM 2 2µ 2 Ze2 Ψ = EΨ, 4πɛ 0 r where 2 CM is the Laplacian for the center-of-mass coordinates R CM = Mr a + mr e M + m 2 is the Laplacian for the coordinates of the electron relative to the nucleus: r = r e r a µ is the reduced mass µ = (M 1 + m 1 ) 1 m. Jay Taylor (ASU) APM Lecture 3 Fall / 56

6 Hydrogen-like atom Reduction of Dimension Because the potential energy depends only on the distance of the electron to the nucleus, the center-of-mass can be separated out of the previous equation, leaving 2 2µ 2 Ψ Ze2 4πɛ 0 r Ψ = EΨ where r = r. Transforming to polar coordinates (r, θ, φ) gives [ 2 1 2m r 2 sin(θ) ( r 2 ) + ( sin(θ) ) + 1 sin(θ) r r θ θ sin(θ) 2 φ 2 ] Ψ Ze2 4πɛ 0 r Ψ = EΨ. Jay Taylor (ASU) APM Lecture 3 Fall / 56

7 Hydrogen-like atom Separation of Variables The last equation can be solved by separation-of-variables and leads to the following set of eigenfunctions and eigenvalues which are indexed by Ψ nlm (r, θ, φ) = C nl R nl (r)yl m (θ, φ) ( ) Ze 2 2 µ E nlm = 2ɛ 0 h 2n 2, the principal quantum number n = 1, 2, 3, ; the azimuthal quantum number l = 0, 1, 2,, n 1; the magnetic quantum number m = l, l + 1,, l 1, l. Jay Taylor (ASU) APM Lecture 3 Fall / 56

8 Hydrogen-like atom Radial Component The radial part of the wavefunction is where R nl (r) = ( 2 r n ) l L 2l+1 n+l ( ) 2 r e r/n n the associated Laguerre polynomial L β α(x) is defined by r = (µe 2 Z/mɛ 0 h 2 )r; L β α(x) = d β dx β L α(x) L α (x) = e x d α [ x α dx α e x] C nl is a normalization constant chosen so that Ψ nlm 2 dr = 1. Jay Taylor (ASU) APM Lecture 3 Fall / 56

9 Hydrogen-like atom Angular Component The angular part of the wavefunction is a spherical harmonic: Yl m (θ, φ) = P m l (cos(θ)) 1 2π eimφ, where the associated Legendre polynomial P m l (x) is defined by Pl m (x) = (1 x 2 m /2 d m ) dx m P l(x) d l P l (x) = 1 [ (x 2 2 l l! dx l 1) l]. Notice that, in general, this function is complex except when l = 0. Jay Taylor (ASU) APM Lecture 3 Fall / 56

10 Hydrogen-like atom s-orbitals When l = 0, we must also have m = 0. In this case, the wavefunction Ψ n,0,0 is radially-symmetric and is said to represent the ns orbital. Notice that as n increases: the energy increases; the diffuseness of the orbital also increases; nodes appear. Jay Taylor (ASU) APM Lecture 3 Fall / 56

11 Hydrogen-like atom p-orbitals For larger values of l, radial symmetry is lost. For example, if l = 1, then m = 1, 0, 1 and there are three p orbitals that can be aligned along the coordinate axes. If n = 2, then these have the following appearance: Jay Taylor (ASU) APM Lecture 3 Fall / 56

12 Separation of timescales Born-Oppenheimer Approximation Since analytical solutions are usually unavailable, we try to simplify the problem. The Born-Oppenheimer approximation assumes that: the nuclei are fixed on the timescale of electronic motion (since nuclei are much heavier than electrons); we can treat the electronic wavefunction and energy as functions of the nuclear coordinates R; the total wavefunction can be factored as Ψ tot (r, R) = Ψ el (r, R)Ψ nuc (R). This leads to the electronic Schrödinger equation: H el (R)Ψ el (r, R) = E el (R)Ψ el (r, R). Jay Taylor (ASU) APM Lecture 3 Fall / 56

13 Separation of timescales The Electronic Hamiltonian The electronic Hamiltonian H el is equal to the sum of the electronic kinetic energy and the coulomb potential energy: T el = 2 elec ( 2 2m x 2 i i V el 1 elec nuc = 4πɛ 0 i s + 2 yi zi 2 Z s e 2 elec + r is i<j where Z i is the atomic number of the i th nucleus. ) e 2 r ij Jay Taylor (ASU) APM Lecture 3 Fall / 56

14 Separation of timescales Effective Potential Energy Surface Typically, we are interested in the electronic ground state energy, E el 0 (R), which can be used to estimate the potential energy surface for the nuclear configuration: This can be used to: E nuc (R) E el 0 (R) + 1 4πɛ 0 nuc s<t Z s Z t e 2 R st identify the minimum energy molecular structure; characterize reaction trajectories; solve for the nuclear wavefunction: H nuc Ψ nuc (R) = E tot Ψ nuc (R). However, even with this approximation, the Schrödinger equation still must be solved numerically. Jay Taylor (ASU) APM Lecture 3 Fall / 56

15 Molecular orbital theory Molecular Orbital Theory Molecular orbital theory attempts to approximate the full wavefunction Ψ using a collection of one electron functions called spin orbitals χ(x, y, z, ξ). These usually have the form ψ(x, y, z)α(ξ) or ψ(x, y, z)β(ξ) where ξ denotes the spin angular momentum of the electron along the z-axis and the spin wavefunctions for spin-up and spin-down particles are ( α + 1 ) 2 ( β + 1 ) 2 ( = 1, α = 0, β 1 ) = 0 2 ) = 1. ( 1 2 Jay Taylor (ASU) APM Lecture 3 Fall / 56

16 Molecular orbital theory Fermi-Dirac Statistics Because electrons have half-integer spin, the following two conditions must be satisfied: The wavefunction must be antisymmetric: if π ij (X) denotes the configuration obtained by permuting the positions and spins of the i th and j th electrons, then Ψ(π ij (X)) = Ψ(X). Pauli exclusion principle: no two electrons can occupy the same spin orbital function. Thus each spatial orbital function can contain at most two electrons, one with spin +1/2 and the other with spin 1/2. Jay Taylor (ASU) APM Lecture 3 Fall / 56

17 Molecular orbital theory Slater Determinants Given a collection of linearly independent spin orbitals, χ 1,, χ n, an antisymmetric n-electron wavefunction can be constructed by setting Ψ det = = 1 n! χ 1 (X 1 ) χ 2 (X 1 ) χ n (X 1 ) χ 1 (X 2 ) χ 2 (X 2 ) χ n (X 2 )... χ 1 (X n ) χ 2 (X n ) χ n (X n ) n ( 1) π χ i (X π(i) ) π S n i=1 1 n! where X i = (x i, y i, z i, ξ i ) denotes the location and spin of the i th electron. Ψ det is called a Slater determinant. Jay Taylor (ASU) APM Lecture 3 Fall / 56

18 Molecular orbital theory Basis Set Expansions In practice, the molecular orbitals ψ i (x) are expressed as linear combinations of one-electron functions known as basis functions, ψ i (x) = N c si φ s (x), 1 i N, s=1 where the c si are the molecular orbital expansion coefficients. Notice that 2N n, if we allow for double occupancy of orbitals; if 2N > n, then there are unoccupied orbitals that do not appear in the Slater determinant. Jay Taylor (ASU) APM Lecture 3 Fall / 56

19 Molecular orbital theory Linear Combination of Atomic Orbitals Usually, the basis elements φ s are taken to be atomic orbitals associated with individual nuclei present in the molecule (LCAO method). In this case, a set of orbitals is assigned to each nucleus such that these orbitals are centered at the nucleus; depend on the type of nucleus (e.g., H vs. C); allow for symmetric and polarized electron distributions. It is often the case that several such basis elements are used to represent an actual atomic orbital, for example, by allowing for different degrees of diffuseness of the orbital about the nucleus. Jay Taylor (ASU) APM Lecture 3 Fall / 56

20 Molecular orbital theory Slater-type atomic orbitals One choice for the basis functions are the Slater-type orbitals (STOs): φ 1s = φ 2s = φ 2px = ( ) ζ 3 1/2 1 exp( ζ 1 r) π ( ) ζ 5 1/2 2 r exp( ζ 2 r/2) 96π ( ) ζ 5 1/2 2 x exp( ζ 2 r/2) 32π. where ζ 1, ζ 2, are parameters that control the size of the orbitals. While STOs are good approximations for atomic orbitals, integrals of their products must be evaluated numerically. Jay Taylor (ASU) APM Lecture 3 Fall / 56

21 Molecular orbital theory Gaussian-type atomic orbitals Gaussian orbital functions are polynomials multiplied by exp( αr 2 ): g s = g x = g y = ( ) 2α 3/4 exp( αr 2 ) π ( ) 128α 5 1/2 x exp( αr 2 ) π 3 ( ) 128α 5 1/2 y exp( αr 2 ) π 3. Although integrals involving Gaussian-type functions can be evaluated explicitly, these functions are poor approximations for atomic orbitals. Jay Taylor (ASU) APM Lecture 3 Fall / 56

22 Molecular orbital theory Contracted Gaussians A compromise can be reached by using linear combinations of Gaussian functions (called contracted Gaussians): φ s (x) = α d sα g α (x) where each g α (x) is a Gaussian function and the coefficients d sα are fixed in advance, i.e., do not depend on the Hamiltonian. One approach is to choose the coefficients to minimize the least squares distance between the contracted Gaussian and a STO: ɛ nl = ( φ STO nl ) 2 φ CG nl dx. Jay Taylor (ASU) APM Lecture 3 Fall / 56

23 Hartree-Fock theory Hartree-Fock Theory Having chosen a set of basis functions, Hartree-Fock theory seeks to estimate the ground state energy E 0 by finding a determinantal wavefunction Φ that minimizes the quantity: E = Φ HΦdX E 0. This leads to variational conditions on the expansion coefficients E c ri = 0 1 r, i N. Thus, we have reduced an infinite-dimensional linear problem to a finite-dimensional non-linear one (see below). Jay Taylor (ASU) APM Lecture 3 Fall / 56

24 Hartree-Fock theory Roothan-Hall Equations For a closed-shell system (i.e., two electrons per orbital), the variational conditions lead to a system of non-linear equations N (F rs ɛ i S rs )c si = 0 1 r, i N, s=1 where ɛ i is the one-electron energy of molecular orbital ψ i and S rs is the overlap of the atomic orbitals φ r and φ s : ɛ i = ψi (x)hψ i (x)dx S rs = φ r (x)φ s (x)dx. Jay Taylor (ASU) APM Lecture 3 Fall / 56

25 Hartree-Fock theory The Fock matrix The Fock matrix F = (F rs ) is defined as F rs = h rs + AO p,q P qp [ (rp sq) 1 2 (rp qs) ] where the summation goes over atomic orbitals and h rs is a component of the one-electron energy in a field of bare nuclei: h rs = φ r (x)ĥ(1)φ s (x)dx ĥ(1) = 2 2m 2 1 Nuc Z a. 4πɛ 0 r a a1 Jay Taylor (ASU) APM Lecture 3 Fall / 56

26 Hartree-Fock theory Bond-order Matrix The matrix P = (P pq ) is called the bond-order matrix or the one-electron density matrix: MO P qp = 2 cpic qi, where the summation is over occupied molecular orbitals and the factor of 2 reflects the closed-shell assumption. i Jay Taylor (ASU) APM Lecture 3 Fall / 56

27 Hartree-Fock theory Two-electron Integrals Much of the computational burden in the H-F theory comes from the need to compute the two-electron repulsion integrals ( ) 1 (rp sq) = φ r (x 1 )φ p(x 2 ) φ s (x 1 )φ q (x 2 )dx 1 dx 2, where r 12 = x 2 x 1. Because the elements of the Fock matrix are themselves functions of the molecular orbital expansion coefficients, the Roothan-Hall equations are non-linear and must be solved iteratively (e.g., Newton-type methods). r 12 Jay Taylor (ASU) APM Lecture 3 Fall / 56

28 Hartree-Fock theory Self-Consistent Field Method A simple iterative scheme for solving for the coefficient matrix c is: 1 Use c (n) to form the bond-order matrix P (n). 2 Use P (n) to form the Fock matrix F (n). 3 Find c (n+1) and ɛ (n+1) by solving the secular equation ( ) F (n) ɛ (n+1) S c (n+1) = 0. 4 Choose the N/2 molecular orbitals of lowest energy to be occupied. 5 Repeat steps (1)-(4) until c (n+1) c (n) < δ. Jay Taylor (ASU) APM Lecture 3 Fall / 56

29 Electron correlation Electron Correlation There are two sources of negative correlation between the locations of different electrons. The Coulomb hole results from electrostatic repulsion. The exchange hole is a consequence of the Pauli exclusion principle: electrons with the same spin cannot occupy the same orbital. The most important limitation of the H-F method is that it fails to account for the Coulomb hole. This is particularly problematic when modeling large molecules, for which about half of the interaction energy can be due to electron correlation. Jay Taylor (ASU) APM Lecture 3 Fall / 56

30 Electron correlation Hartree-Fock wavefunction of H 2 Example: The H-F wave function for the hydrogen molecule H 2 with double occupancy of a single molecular orbital φ( ) is ψ(1, 2) = 1 χ 1 (1) χ 2 (1) 2 χ 1 (2) χ 2 (2) = 1 φ(x 1 )φ(x 2 ) [ α(ξ 1 )β(ξ 2 ) β(ξ 1 )α(ξ 2 ) ], 2 where (x i, ξ i ) is the location and spin of electron i, ξ 2 = ξ 1, and α( ) and β( ) are the spin-up and spin-down functions introduced earlier. Jay Taylor (ASU) APM Lecture 3 Fall / 56

31 Electron correlation The H-F wave function for H 2 neglects electronic correlation. The joint probability density of the locations of the two electrons is ψ(1, 2) 2 = Cφ 2 (x 1 )φ 2 (x 2 ) where C is a normalizing constant. Since this density factors into the product of two one-electron densities, it follows that under the H-F wave function, the locations of the two electrons are independent of one another. This result is unphysical, but is also observed in H-F calculations for more complicated molecules. Jay Taylor (ASU) APM Lecture 3 Fall / 56

32 Electron correlation Post-Hartree-Fock Methods Several methods have been devised to better account for electronic correlation: Configuration Interaction (CI) method Coupled Cluster (CC) method. Many Body Perturbation Theory (MBPT), including the Møller-Plesset perturbation theory. However, these are even more computationally intensive than the H-F method. Jay Taylor (ASU) APM Lecture 3 Fall / 56

33 Electron correlation Occupancy Given a collection of spin-orbitals, χ 1, χ 2, and a Slater determinant Ψ, we define the occupancy of spin-orbital χ i to be { 1 if χi appears in Ψ n i = 0 otherwise, and we write Ψ = Ψ n (n 1, n 2, ). Here we require that n k = n. k=1 Jay Taylor (ASU) APM Lecture 3 Fall / 56

34 Electron correlation Creation and Annihilation Operators The creation and annihilation operators are defined as: ˆk Ψ n ( n k ) = θ k (1 n k )Ψ n+1 ( 1 n k ), ˆkΨ n ( n k ) = θ k n k Ψ n 1 ( 1 n k ), where θ k = ( 1) P j<k n j. Thus ˆk creates an electron and places it in the unoccupied spin-orbital χ k, while ˆk annihilates the electron in the occupied spin-orbital χ k. Jay Taylor (ASU) APM Lecture 3 Fall / 56

35 Electron correlation Configuration Interaction In the CI method, the H-F wave function is replaced by a linear combination of Slater determinants formed from different sets of n spin-orbitals: Ψ CI = c 0 Ψ 0 + a,p c a p ˆp âψ 0 + a<b,p<q c ab pq ˆq ˆp âˆbψ 0 +. Here, Ψ 0 is usually the H-F wave function and the coefficients cp, a are chosen to minimize the energy: E CI = Ψ CI HΨ CI dx Ψ 0HΨ 0 dx E HF. Jay Taylor (ASU) APM Lecture 3 Fall / 56

36 Electron correlation Partial Configuration Interaction As a rule, the full CI calculation is not feasible, since ( ) 2N n different n-electron determinants can be formed from a set of N molecular orbitals and we usually take N n. Instead, the CI expansion is usually restricted to lower order terms (called excitations): CID includes all doublet excitations c ab pq ˆq ˆp âˆbψ 0 ; CISD includes all singlet and doublet excitations; frozen shell CI only allows outer shell orbitals to be substituted. Jay Taylor (ASU) APM Lecture 3 Fall / 56

37 Electron correlation Coupled Cluster Method The coupled cluster method seeks to find an operator T such that the exact ground state wave function can be written as Ψ = e T Ψ 0 where Ψ 0 is usually the H-F wave function and the cluster operator T is a sum of excitation operators with T = T 1 + T 2 + T 3 + T 1 = a,r t r aˆr â (single excitations) T 2 = 1 4 a,b,r,s t rs abŝ ˆr âˆb (double excitations). Jay Taylor (ASU) APM Lecture 3 Fall / 56

38 Electron correlation The amplitudes t r a, can be characterized in the following way. First, we write He T Ψ 0 = Ee T Ψ 0 which, upon multiplying both sides by e T, gives e T He T Ψ 0 = EΨ 0. Next, we use the commutator expansion to write e T He T = H + [H, T ] + 1 2! [[H, T ], T ] + 1 [[[H, T ], T ], T ] 3! + 1 [[[[H, T ], T ], T ], T ] 4! where [A, B] AB BA. The key observation is that the commutator expansion terminates because H only has two-particle interactions. Jay Taylor (ASU) APM Lecture 3 Fall / 56

39 Electron correlation Calculating the CC Amplitudes If we then substitute the commutator expansion into (*) and multiply by the function Ψ m 1 m l a 1 a l = ˆm l ˆm 1â1 â l Ψ 0 and finally integrate over the spin and spatial coordinates, then we obtain the identity ( (Ψ ) m 1 m l a 1 a l H + [H, T ] ) 4! [[[[H, T ], T ], T ], T ] Ψ 0 dx = 0. The right-hand side vanishes because Ψ 0 and Ψ m 1 m l a 1 a l are orthogonal. This leads to a system of fourth-order polynomial equations in the CC amplitudes which in principle can be solved iteratively. As with CI, the cluster expansion is usually truncated to obtain CCD or CCSD. Jay Taylor (ASU) APM Lecture 3 Fall / 56

40 Electron correlation Møller-Plesset (MP) Perturbation Theory In the Møller-Plesset perturbation theory, we write the Hamiltonian as H = H (0) + H (1) = ɛ i î î + H (1), i=0 where ɛ i is the one-electron energy of the i th molecular orbital and the H-F wave function Ψ 0 is the ground state eigenfunction for H (0). We can then expand the true ground state wave function and energy as series Ψ 0 = n=0 Ψ (n) 0, E 0 = n=0 E (k) 0 where the terms in these series can be evaluated recursively, usually up to second (MP2) or sometimes fourth (MP4) order. Jay Taylor (ASU) APM Lecture 3 Fall / 56

41 Electron correlation More formally, if Ψ (0) k and E (0) k form a complete set of eigenvalues and eigenfunctions for H (0), then we can define the reduced resolvent ( ) 1 R 0 = n=1 E (0) 0 E n (0) P n. where P n is the orthogonal projection onto Ψ (0) n. It can then be shown that Ψ 0 = n=0 S n Ψ (0) 0 E 0 = E (0) 0 + Ψ (0) 0, H(1) n=0 S n Ψ (0) 0, where S R 0 (E (0) 0 E 0 + H (1) ). The difficulty here is that S also depends on E 0. Jay Taylor (ASU) APM Lecture 3 Fall / 56

42 Applications Šponer et al. (2004). Accurate Interaction Energies of Hydrogen-Bonded Nucleic Acid Base Pairs. JACS 126: Background Base pair interactions in DNA and RNA depend largely on H-bonding. Experimental measurement of the H-bond energies is difficult. This motivates ab initio calculations of these energies. Jay Taylor (ASU) APM Lecture 3 Fall / 56

43 Applications Interaction Energies The interaction energy of dimer A B is defined as where E A B = E A B (E A + E B ) + E Def E A B is the energy of the optimized dimer; E A, E B are the energies of the isolated bases, with the geometries of the optimized dimer, calculated using the dimer basis set; E Def is the deformation energy of the two isolated bases. Thus, the interaction energy is equal to the energy of the H-bonds less the energy required to deform the geometries of the isolated bases. Jay Taylor (ASU) APM Lecture 3 Fall / 56

44 Applications Deformation Energy The deformation energy of dimer A B is defined as where E Def = (E A E A mon) + (E B E B mon) E A, E B are the energies of the isolated bases, with the geometries of the optimized dimer, calculated using the monomer basis sets Emon, A E B mon are the energies of the isolated bases, with the geometries optimized in isolation using the monomer basis sets. Here the monomer basis sets are used in both calculations to avoid basis set superposition error (BSSE). Jay Taylor (ASU) APM Lecture 3 Fall / 56

45 Applications Extrapolated Energies Extrapolation of energies to the complete basis set (CBS) limit was done using Helgaker s formula where E corr CBS E corr X is the extrapolated energy; = ECBS corr 3 + BX X is the number of basis functions used to represent each valence orbital (X = 2, 3, 4); EX corr is the calculated energy, using the MP2 perturbation theory with an aug-cc-pvxz basis set; The adz atz values reported in Table 2 were obtained by extrapolating from X = 2 and X = 3. Jay Taylor (ASU) APM Lecture 3 Fall / 56

46 Applications Results dimer E A B E SCF E corr E Def AMBER GC (WC) AT (WC) GU (wobble) GA AA AMBER is a molecular mechanics package that uses a particular set of force fields. Jay Taylor (ASU) APM Lecture 3 Fall / 56

47 Applications Conclusions Comparison with higher-order extrapolations suggests that the adz atz are within 1 kcal/mol of the true values. Electron correlation (dispersion attraction + intramolecular correlation) contributes significantly to the interaction energies. H-bond energies estimated using AMBER are within 3 kcal/mol of the the QM estimates, with greater discrepancies for weak base pairs. Base pair stability is mainly determined by electrostatic interactions that can be approximated by atom-centered charges. Jay Taylor (ASU) APM Lecture 3 Fall / 56

48 Applications Schreier et al. (2007). Thymine Dimerization in DNA Is an Ultrafast Photoreaction. Science 315: Boggio-Pasqua et al. (2007). Ultrafast Deactivation Channel for Thymine Dimerization. JACS 129: Background Thymine dimerization occurs through UV irradiation of DNA sequences containing adjacent thymine bases (TT dinucleotides). Thymine dimers are usually repaired by photoreactivation or by nucleotide excision pathways. Unrepaired dimers are mutagenic and are believed to be a major contributor to melanoma. Jay Taylor (ASU) APM Lecture 3 Fall / 56

49 Applications Photocycloaddition Absorption of a photon by thymine excites a π electron to a π orbital. This can then: Return to the ground state (S 0 ) via internal conversion of the excitation energy to heat. Attack the double bond on an adjacent thymine, leading to dimerization. Thermal activation of the cycloaddition reaction has low yield because of improper symmetry of the interacting orbitals. Jay Taylor (ASU) APM Lecture 3 Fall / 56

50 Applications Femtosecond Time-Resolved IR Spectroscopy Schreier et al. (2007) studied thymine dimerization in UV-irradiated (dt ) 18 using IR spectroscopy. This indicates that TD formation occurs within 3 ps of UV absorption. Jay Taylor (ASU) APM Lecture 3 Fall / 56

51 Applications Quantum Mechanical Analysis of Thymine Dimerization Boggio-Pasqua et al. (2007) used molecular orbital theory to characterize the likely reaction pathways for thermal and photochemical thymine dimerization. The QM calculations used multi-configurational SCF (CASSCF) and perturbation theory (CASPT2) to calculate energies of intermediate states. These methods allow for electron excitation to higher-energy orbitals. The potential energy surface near the UV-excited state has the geometry of a conical intersection (CI). Jay Taylor (ASU) APM Lecture 3 Fall / 56

52 Applications Thermally-driven TD formation has low yield. The thermal reaction pathway passes through two transition states to lead to a thymine dimer that is higher in energy than the TT dinucleotide. Thymine dimerization is unlikely to occur through this mechanism. Jay Taylor (ASU) APM Lecture 3 Fall / 56

53 Applications The photochemical reaction is barrierless. In contrast, following photon absorption, the excited system will relax to a CI from which it can either rapidly relax to the thymine dimer or return to the TT ground state. Jay Taylor (ASU) APM Lecture 3 Fall / 56

54 Applications The conical intersection. Relaxation of the excited state occurs so rapidly that the neighboring dimers must have a suitable conformation at the time of photoexcitation for thymine dimerization to occur. Jay Taylor (ASU) APM Lecture 3 Fall / 56

55 Summary Quantum Chemistry: Scope and Limitations Quantum mechanical calculations are important whenever: Empirical data relevant to molecular energetics are lacking or have questionable accuracy; We wish to study processes involving bond formation and breaking (chemical reactions). However, all of these methods are computationally intensive and so are usually restricted to aperiodic systems containing at most a tens of atoms. Jay Taylor (ASU) APM Lecture 3 Fall / 56

56 Summary References Hehre, W. J., Radom, L., Schleyer, P. v. R., and Pople, J. A. (1986) Ab Initio Molecular Orbital Theory. Wiley. Piela, L. (2007) Ideas of Quantum Chemistry. Elsevier. Jay Taylor (ASU) APM Lecture 3 Fall / 56