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Diagonalization, Eigenvalues Problem, Secular Equation Diagonalization, eigenvalues problem, secular equation Diagonalization procedure of a matrix A with dimensionality n × n (e.g. the Hessian matrix) 2 3 a11 a12 : : : a1n 6 7 6 a21 a22 : : : a2n 7 6 7 A = 6 . 7 6 . 7 4 5 an1 an2 : : : ann consists in finding a matrix C such, that the matrix D=C−1AC is diagonal: 2 3 d1 0 ::: 0 6 7 6 0 d2 ::: 0 7 −1 6 7 C AC = D = 6 . 7 6 . 7 4 5 0 0 : : : dn Equation C−1AC=D can also be written as AC = CD Denoting the elements of matrix C by cij and using te fact that D is diagonal (i.e., its elements dij are of the form djδij, where δij denotes the Kronecker delta, δii=1 i δij=0 for i6=j) we obtain X X X (AC)ij = aik ckj (CD)ij = cik dkj = cik dj δkj = djcij k k k Diagonalization, eigenvalues problem, secular equation Employing the equation (AC)ij = (CD)ij we obtain X aik ckj = djcij k In matrix notation this equation takes the form ACj = dj Cj where Cj is the jth column of matrix C. This is equation for the eigenvalues (dj) and eigenvectors (Cj) of matrix A. Solving this equation, that is the solving the so called eigenproblem for matrix A, is equivalent to diagonalization of matrix A. This is because the matrix C is built from the (column) eigenvectors C1;C2;:::;Cn: C = [C1;C2;:::;Cn] The equation for eigenvectors can also be written as( A − dj E) Cj = 0, where E is the unit matrix with elements δij. This equation has a solution only if the determinant of the matrix A − dj E vanishes jA − djEj = 0 This is the very important and practically useful secular equation for eigenvaules dj. Hartree-Fock Theory The wave function Φ in the Hartree-Fock theory is a Slater determinant φ (1) φ (2) : : : φ (N) 1 1 1 1 φ2(1) φ2(2) : : : φ2(N) Φ(1; 2;:::;N) = p . N! . φN (1) φN (2) : : : φN (N) built from molecular spinorbitals φk(i)=φk(~ri; σi)=φk(xi; yi; zi; σi). Symbolically: Φ =j φ1 φ2 φ3 : : : φN j Molecular spinorbitals φi are determined by minimalization of the energy functional: Z E[Φ] = Φ∗H^ Φdτ where H^ is the electronic Hamiltonian of a molecule. International Academy of Quantum Molecular Science http://www.iaqms.org/deceased/fock.php VLADIMIR A. FOCK Born on December 22, 1898. Professor of Physics, University of Leningrad (U.R.S.S.). Member of the Academy of Sciences of the Soviet Union. Member of the Academies of Sciences of Germany, Norway and Denmark. Author of: "Principles of wave-mechanics", "The theory of Space, Time and Gravitation", "Electromagnetic Diffraction and Propagation Problems". Important Contributions: Relativistic equation of wave-mechanics (Klein–Fock equation, 1926). Dirac equation and Riemann goemetry (1929). Hartree–Fock method (1930). Quadridimentionnal symmetry of hydrogen atom (1935). Quantum field theory (Fock space, Fock representation 1932–1937). Gravitation theory (1939–1950). Propagation of electromagnetic waves (1944–1965). Epistemological research on the theory of relativity (relativity principle with respect to observation procedure (1949–1973). 1 z 1 2010-11-14 19:56 Hartree-Fock Theory, continued • RHF (Restricted Hartree-Fock) Method For 6 electrons it is sufficient to use 3 or 4 orbitals: 1, 2, 3, i 4 Singlet states: Φ=j 1α 1β 2α 2β 3α 3β j Triplet states: Φ=j 1α 1β 2α 2β 3α 4α j (because of double occupancy of orbitals). • UHF (Unrestricted Hartree-Fock) Method For 6 electrons one has to use 6 orbitals: 0 0 0 Singlet state: Φ=j 1α 1β 2α 2β 3α 3β j 0 0 Triplet state: Φ=j 1α 1β 2α 2β 3α 4α j (because different orbitals for different spins are used) Both methods are employed in practice. Both have advantages and disadvantages. The RHF method is simpler, gives a state of a well defined, pure spin, but fails to correctly describe chemical bond dissociation. The UHF is more time-consuming, correctly describes chemical bond dissociation, but gives states of undefined spin (spin contamination) and (often) artifacts on potential energy surfaces. Hartree-Fock equations fφ^ k = "kφk where "k is the so-called orbital energy and f^ is the Fock operator f^ = h^ + J^ − K^ h^ denotes the sum of the kinetic energy operator and the attractive nuclear potential: 1 X Zj h^ = − ∆ − 2 ~ j j~r − Rjj The Coulomb J^ and exchange K^ operators are more complicated. The Coulomb operator J^ depends linearly on the electron density ρ(~r) defined as : occ X 2 ρ(~r; σ) = φk(~r; σ) k In particular J^ represents the multiplication by the averaged potential j(~r) of the electron cloud: Z 1 j(~r) = ρ(~r 0)d ~r 0 (1) j~r − ~r 0j Hartree-Fock equations, continued The exchange operator K^ is more complicated. This is an integral operator depending on all occupied orbitals: occ Z X 0 1 0 0 (K^ )(~r) = φk(~r ) (~r )d ~r φk(~r) (2) j~r − ~r 0j k (you do not have to memoraize this formula). The Hartree-Fock energy EHF is computed in the following way: Z X 1 X EHF = ΦH^ Φdτ = "k − (Jkk − Kkk) (3) 2 i k where Z Z Jkk = φkJφ^ kdτ Kkk = φkKφ^ kdτ (4) are the Coulomb and exchange integrals, respectively. Very important in applications is the Koopmans theorem concerning the ionization potential (IP) and the electron affinity (EA) of an atom or a molecule: IP. = −"HOMO EA = "LUMO where HOMO denotes the highest occupied and LUMO the lowest unoccupied molecular orbital (MO). LCAO MO Method. Roothaan Equations In the LCAO MO method the molecular spinorbitals are represented as linear combinations of atomic spinorbitals χj(~r; σ): 2M X φk(~r; σ) = Cjk χj(~r; σ) j=1 Spinorbitals χj are not exact atomic spinorbitals but to a large extent arbitrary functions (basis functions) localized on atomic centers in a molecule. In particular, the functions χj are expressed through M functions (orbitals) of an atomic basis bj(~r): χ2j−1(~r; σ) = bj(~r)α(σ) χ2j(~r; σ) = bj(~r) β(σ) The choice of the atomic basis bj(~r) and ist size M determine the accuracy of calculations. The linear coefficients Cjk and orbital energies "k are found by solving the Roothaan equations: FCk = "kSCk where F is the Fock matrix, S is the overlap matrix , and Ck is the kth column of matrix C Z Z ∗ ^ ∗ Fij = χi fχjdτ Sij = χi χjdτ International Academy of Quantum Molecular Science http://www.iaqms.org/members/roothaan.php CLEMENS C. J. ROOTHAAN Born August 29, 1918 in Nymegen, Netherlands. Louis Block Professor of Physics and Chemistry, Emeritus, University of Chicago, Illinois, USA. Email: [email protected] Clemens Roothaan was educated at the Technical Institute Delft (MS 1945) and at the University of Chicago (PhD, Physics 1950). He was Research Associate (1949–50), Instructor to Professor of Physics and Chemistry (1950–), Professor of Communications and Information Science (1965–68), Director of the Computer Center (1962–68), at the University of Chicago. He was also Guggenheim Fellow, Cambridge University (1957), Consultant for : Argonne National Laboratory (1958-66), Lockheed Missiles and Space Company (1960-65), Union Carbide Corporation (1965-), IBM Corporation (1965-). He has been Visiting Professor at the Ohio State University (1976), the Technical University, Lyngby, Denmark (1983), University of Delft (1987-88). He is a Fellow of the American Physical Society and a Corresponding Member of the Royal Dutch Academy of Science. Important Contributions: The theory of atomic and molecular structure. Application of digital computers to scientific problems. 1 z 1 2010-11-14 20:06 The SCF Method In practice the Roothaan equations FCk = "kSCk are solved iteratively. In the nth iteration we diagonalize the Fock matrix F computed using the orbitals from the (n − 1)th iteration. The most time-consuming step is the calculation of M 4=8 two-electron integrals (including four-center ones): ZZ ∗ ∗ 1 hpqjrsi = bp(~r1) bq(~r2) br(~r1) bs (~r2) dτ1dτ2 j~r1 − ~r2j needed to form the Fock matrix F (remember that f^ = h^ + J^ − K^ ). The choice of the atomic basis bj(~r) and ist size M determine the accuracy of calculations. Each kind of an atom requires different basis. Up to now hundreds of basis sets have been developed. Initially Slater basis sets , mainly minimal Slater bases were used: n−l −ζr Snlm(~r) = r e Ylm(θ; φ) For instance, for atoms Li to Ne, the minimal basis (MBS) consists of only 5 functions: −ζr −ζr −ζr −ζr −ζr 1s = e 2s = r e 2px = x e 2py = y e 2px = z e Gaussian Bases In 1950 Frank Boys made a breakthrough discovery. He observed that the product of Gaus- sian functions, e−γr2, localized on different atoms is again a Gaussian function (localized at a point between them). Due to this property all two-electron integrals, including the four-center ones are expressible through very simple, closed form formulas and can be quickly computed. Boys proposed to use in SCF calculations the Gaussian basis functions of the following general form: p q s −γr2 Gpqs(~r) = x y z e In particular, the 1s i 2p Gaussian functions are of the form: −γr2 −γr2 −γr2 −γr2 1s = e 2px = x e 2py = y e 2px = z e There are two kinds of d functions. We use either the 5 spherical Gaussian functions : 2 −γr2 G3d;m(~r) = r e Y2m(θ; φ) or the 6 Cartesian ones 2 −γr2 2 −γr2 2 −γr2 −γr2 −γr2 −γr2 dx2 =x e dy2 =y e dz2 =z e dxy =xye dxz =xze dyz =yze The Gaussian functions of the type 2s, 3p, 4d, etc, (with odd powers of r) are not used.
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