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PCLOBE

PCLOBE. Quantum Chemistry on a PC. Important Lobe-Basis Papers. HISTORICAL FOUNDATIONS OF GAUSSIAN-LOBE BASIS SETS 1. S. F. Boys, Proc. Roy. Soc. A200, 542 (1950) 2. J. L. Whitten, J. Chem. Phys. v39, 349 (1963) 3. I. Shavitt, Methods in Comp. Phys. v2, p1 (1963)

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PCLOBE

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  1. PCLOBE Quantum Chemistry on a PC

  2. Important Lobe-Basis Papers • HISTORICAL FOUNDATIONS OF GAUSSIAN-LOBE BASIS SETS • 1. S. F. Boys, Proc. Roy. Soc. A200, 542 (1950) • 2. J. L. Whitten, J. Chem. Phys. v39, 349 (1963) • 3. I. Shavitt, Methods in Comp. Phys. v2, p1 (1963) • 4. H. Sambe, J. Chem. Phys. v42, 1732 (1965) • 5. H. Preuss and G. Diercksen, Int. J. Quantum Chem. I, 605 (1967) • 6. J. F. Harrison and L. C. Allen, Molec. Spectrosc. v29, 432 (1969) • 7. F. Dreisler and R. Ahlrichs, Chem. Phys. Lett. v23, 571 (1973) • 8. H. Le Rouzo and B. Silvi, Int. J. Quantum Chem. XIII, 297 (1978) • 9. S. Y. Leu and C. Y. Mou, J. Chem. Phys. v101, 5910 (1994) • 10. P. Otto, H. Reif and A. Hernandez-Laguna, • J. Mol. Structure (Theochem) v340, 51 (1995)

  3. Short History of Quantum Chemistry • 1901, M. Planck: E = n hv • 1905, A. Einstein: E = Mc2 • 1913, N. Bohr: mvr = n h/2π • 1923, L. De Broglie: λ = h/mv • 1926, E. Schroedinger: E = -(Z/n)2 (13.605 ev) for H-atom • 1931, P.A.M. Dirac, Relativistic H-atom

  4. Short History of Chemical Bonding • 1927, Heitler and London H-H, Valence Bond • 1928, R. Mulliken H-H, Molecular Orbitals • 1930, E. Huckel, Molecular Orbitals • 1933, L. Pauling, Valence Bond Theory • 1950, S. F. Boys, Gaussian (Lobe) integrals • 1953, Bates, Ledsham and Stewart, H2+, exact • 1960, Boys and Reeves, first CH2O SCF • 1963, Kolos & Wolniewicz, H2 exact • 1970, Pople GAUSSIAN-70

  5. PCLOBE USER INTERFACE

  6. Interface to CHEMSITE • Model of Crown ligand

  7. Interface to MOLUCAD • D-glucopyranose

  8. Basis Sets • The main idea of PCLOBE is to use atomic orbitals formed from linear combinations of off-center gaussian spheres. The s-orbitals are the same as cartesian-gaussians, but p-, d- and f-orbitals are formed from linear combinations of s-gaussians located off the position of the atomic nucleus.

  9. Advantages of a Lobe Basis 1. In development of new applications, integrals only need to be worked out for s-orbitals. 2. The computer program tends to be smaller (less memory) due to the need for only s-orbital routines. 3. Important integrals such as the screened-electron correlation are much simpler in a s-orbital basis, hence easier to compute.

  10. Disadvantages of Lobe Basis Sets 1. There is slight contamination of p-orbitals by f-orbitals and contamination of d-orbitals by s-orbitals. Although it is possible to purify the orbitals there is a practical limit due to numerical precision. 2. Charge-distribution quantities like dipole moments show some asymmetry due to orbital contamination. 3. Orbital integrals have to be summed over many s-orbital components.

  11. Unique Features of PCLOBE • PCLOBE offers several features not available with other programs. • 1. Correlated-SCF calculations for molecules containing H to F. • 2. CD and MCD rotational strengths for UV bands of organic molecules. • 3. Spin-projected open shell calculations for metal complexes (Boys-Reeves C.I.) • 4. Morse-curve anharmonic vibrations. • 5. Slater-Transform-Preuss minimum basis. • 6. Verbose output for teaching quantum chemistry. • 7. Accepts input from CHEMSITE or MOLUCAD.

  12. PCLOBE Standard Features • 1. Geometry optimization using several basis sets, STO-3G, STO-4G*, STP-5/4* minimum sets as well as 6-31G**, 6-311G**, 6-311G**++ and Dunning 10s/7p**. • 2. Boys-Reeves Natural Orbital C.I. with Complete Active Space for singlets, doublets, triplets, quartets, quintets and hextets, neutral species or cations. • 3. Single-Excitation CIS for UV, CD and MCD excited states with oscillator strengths. • 4. Harmonic and Anharmonic vibrational frequencies. • 5.Moller-Plesset Second Order Perturbation (singlets). • 6. Dipole moment, Quadrupole Tensor and Mulliken Analysis. • 7. Output files for color pictures of molecules.

  13. Details of Simple Examples • 1. STO-3G geometry optimization and vibrational frequencies for water. • 2. 6-31G** geometry optimization of water at the Hartree Fock SCF level. • 3. 6-31G** geometry optimization of water using the Correlated-SCF method. • 4. UV, CD (none) and MCD spectra of formaldehyde using STO-4G basis CIS. • 5. MCD spectra of Indole using STO-4G CIS.

  14. STO-3G Test on Water Coordinates and Orbital Scaling • Nuclear Coordinates from Input • Atomic Core X Y Z Basis • 8. 0.319008 -0.256483 0.001576 0 • Z1s= 7.660 Z2s= 2.246 Z2p= 2.250 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • 1. 1.279004 -0.256602 -0.001146 0 • Z1s= 1.240 Z2s= 0.000 Z2p= 0.000 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • 1. 0.070645 0.670830 -0.001148 0 • Z1s= 1.240 Z2s= 0.000 Z2p= 0.000 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • Basis Size = 7 and Number of Spheres = 23 for 10 Electrons

  15. Input Geometry for Water • Distance Matrix in Angstroms • O H H • O 0.0000 0.9600 0.9600 • H 0.9600 0.0000 1.5232 • H 0.9600 1.5232 0.0000 • A-B-C Arcs in Degrees for 3 Atoms • arc( 1, 2, 3)= 37.50 • arc( 1, 3, 2)= 37.50 • arc( 2, 1, 3)= 105.00 • arc( 2, 3, 1)= 37.50 • arc( 3, 1, 2)= 105.00 • arc( 3, 2, 1)= 37.50

  16. Primitive Gaussian Basis • Molecular Orbitals are linear combinations of atomic orbital mimics. • Atomic Orbital mimics are linear combinations of spherical gaussians. • /a,Rx,Ry,Rz) = • N exp(-a( (x-Rx)^2 + (y-Ry)^2 + (z-Rz)^2)) • So aside from the normalization coefficient N, the gaussian is defined by a, Rx, Ry and Rz.

  17. Primitive Lobe Basis Set • Spherical Gaussian Basis Set • No. 1 alpha = 0.27982525E+03 at X= 0.602838 Y= -0.484683 Z= 0.002978 a.u. • No. 2 alpha = 0.42050690E+02 at X= 0.602838 Y= -0.484683 Z= 0.002978 a.u. • No. 3 alpha = 0.90505245E+01 at X= 0.602838 Y= -0.484683 Z= 0.002978 a.u. • No. 4 alpha = 0.40348941E+00 at X= 0.602838 Y= -0.484683 Z= 0.002978 a.u. • No. 5 alpha = 0.20120803E+02 at X= 0.602838 Y= -0.484683 Z= 0.002978 a.u. • No. 6 alpha = 0.66217500E+00 at X= 0.676616 Y= -0.484683 Z= 0.002978 a.u. • No. 7 alpha = 0.28552500E+01 at X= 0.676616 Y= -0.484683 Z= 0.002978 a.u. • No. 8 alpha = 0.66217500E+00 at X= 0.529060 Y= -0.484683 Z= 0.002978 a.u. • No. 9 alpha = 0.28552500E+01 at X= 0.529060 Y= -0.484683 Z= 0.002978 a.u. • No. 10 alpha = 0.66217500E+00 at X= 0.602838 Y= -0.410905 Z= 0.002978 a.u. • No. 11 alpha = 0.28552500E+01 at X= 0.602838 Y= -0.410905 Z= 0.002978 a.u. • No. 12 alpha = 0.66217500E+00 at X= 0.602838 Y= -0.558460 Z= 0.002978 a.u. • No. 13 alpha = 0.28552500E+01 at X= 0.602838 Y= -0.558460 Z= 0.002978 a.u. • No. 14 alpha = 0.66217500E+00 at X= 0.602838 Y= -0.484683 Z= 0.076756 a.u. • No. 15 alpha = 0.28552500E+01 at X= 0.602838 Y= -0.484683 Z= 0.076756 a.u. • No. 16 alpha = 0.66217500E+00 at X= 0.602838 Y= -0.484683 Z= -0.070800 a.u. • No. 17 alpha = 0.28552500E+01 at X= 0.602838 Y= -0.484683 Z= -0.070800 a.u. • No. 18 alpha = 0.73328489E+01 at X= 2.416967 Y= -0.484908 Z= -0.002166 a.u. • No. 19 alpha = 0.11019426E+01 at X= 2.416967 Y= -0.484908 Z= -0.002166 a.u. • No. 20 alpha = 0.23716991E+00 at X= 2.416967 Y= -0.484908 Z= -0.002166 a.u. • No. 21 alpha = 0.73328489E+01 at X= 0.133500 Y= 1.267685 Z= -0.002169 a.u. • No. 22 alpha = 0.11019426E+01 at X= 0.133500 Y= 1.267685 Z= -0.002169 a.u. • No. 23 alpha = 0.23716991E+00 at X= 0.133500 Y= 1.267685 Z= -0.002169 a.u.

  18. Contracted Water Orbitals • Contracted Orbital No. 1 • 3.35975482*( 1), 4.62385931*( 2), 2.47553321*( 3), • Contracted Orbital No. 2 • 0.36313765*( 4), -0.36313765*( 5), • Contracted Orbital No. 3 • 3.43658567*( 6), 2.04334155*( 7), -3.43658567*( 8), -2.04334155*( 9), • Contracted Orbital No. 4 • 3.43658567*( 10), 2.04334155*( 11), -3.43658567*( 12), -2.04334155*( 13), • Contracted Orbital No. 5 • 3.43658567*( 14), 2.04334155*( 15), -3.43658567*( 16), -2.04334155*( 17), • Contracted Orbital No. 6 • 0.21882501*( 18), 0.30115771*( 19), 0.16123456*( 20), • Contracted Orbital No. 7 • 0.21882501*( 21), 0.30115771*( 22), 0.16123456*( 23), • ***** Nuclear Repulsion Energy in au = 9.16702063287864 *****

  19. Initial Core Molecular Orbitals for Water • Initial-Guess-Eigenvectors by Column • # At-Orb 1 2 3 4 5 6 7 • 1 O 1s 0.971 0.000 0.000 0.000 0.260 0.000 -0.132 • 2 O 2s 0.175 -0.001 0.001 0.000 -0.985 0.000 0.966 • 3 O 2px 0.018 0.999 -0.001 0.000 0.006 -0.432 0.251 • 4 O 2py 0.023 0.000 -0.999 0.000 0.007 0.332 0.328 • 5 O 2pz 0.000 0.000 0.000 1.000 0.000 0.00 -0.002 • 6 H 1s -0.065 0.002 0.000 0.000 -0.020 0.923 -0.911 • 7 H 1s -0.065 -0.001 -0.003 0.000 -0.020 -0.923 -0.910 • One-Electron Energy Levels • E( 1) = -24.983054728665 • E( 2) = -5.648104909462 • E( 3) = -5.635229585761 • E( 4) = -5.545426743140 • E( 5) = -4.737369927513 • E( 6) = -2.945239464318 • E( 7) = -2.613131660410

  20. Hartree Fock Roothaan Equations • 1. Assume a determinental wavefunction built from n molecular orbitals. • 2. Calculate Core Energy Matrix H = T + V • 3. Solve HC=ESC for the initial C matrix • 4. Calculate Pij = Σk Cki Ckj for k occupied orbitals. • 5.Set up Fij = Hij + ΣkΣl Pkl [(I,j/k,l) –(0.5)(ik,jl)] • 6. Solve FC=ESC for C • 7. Calculate E = (1/2) ΣkΣl Pkl (Hkl + Fkl ) • 8. If E des not converge go back to 4.

  21. SCF Iteration for Water • ( 1, 7/ 7, 7) are done. • ( 2, 7/ 7, 7) are done. • ( 3, 7/ 7, 7) are done. • ( 4, 7/ 7, 7) are done. • ( 5, 7/ 7, 7) are done. • ( 6, 7/ 7, 7) are done. • ( 7, 7/ 7, 7) are done. • Block No. 1 Transferred to Disk/Memory • The Two-Electron Integrals Have Been Computed • Electronic Energy= -82.1807560770 a.u., Dif.= 82.1807560770 • Electronic Energy= -82.2648927964 a.u., Dif.= 0.0841367193 • Electronic Energy= -83.0815660421 a.u., Dif.= 0.8166732458 • Electronic Energy= -83.1278296576 a.u., Dif.= 0.0462636154 • Electronic Energy= -83.1722112188 a.u., Dif.= 0.0443815612 • Electronic Energy= -83.2147633318 a.u., Dif.= 0.0425521130 • Electronic Energy= -83.2555399690 a.u., Dif.= 0.0407766372 • Electronic Energy= -83.2945960169 a.u., Dif.= 0.0390560478 • Electronic Energy= -83.3319868820 a.u., Dif.= 0.0373908652 • Electronic Energy= -83.3677681514 a.u., Dif.= 0.0357812694 • Electronic Energy= -83.4019952988 a.u., Dif.= 0.0342271474

  22. SCF Convergence for Water • alpha = 0.900000 • Electronic Energy= -84.0966488804 a.u., Dif.= 0.0000008639 • Iteration No. 48 Energy Second Derivative =-0.00000004571814 • alpha = 0.900000 • Electronic Energy= -84.0966489829 a.u., Dif.= 0.0000001025 • Iteration No. 49 Energy Second Derivative =-0.00000001042119 • alpha = 0.900000 • Electronic Energy= -84.0966489969 a.u., Dif.= 0.0000000139 • Iteration No. 50 Energy Second Derivative =-0.00000000238134 • alpha = 0.900000 • Electronic Energy= -84.0966489991 a.u., Dif.= 0.0000000022 • Iteration No. 51 Energy Second Derivative =-0.00000000054465 • alpha = 0.900000 • Electronic Energy= -84.0966489995 a.u., Dif.= 0.0000000004 • Iteration No. 52 Energy Second Derivative =-0.00000000012471 • alpha = 0.900000 • Electronic Energy= -84.0966489996 a.u., Dif.= 0.0000000001 • Iteration No. 53 Energy Second Derivative =-0.00000000002848 • alpha = 0.900000 • Electronic Energy= -84.0966489996 a.u., Dif.= 0.0000000000 • ##### Total Energy = -74.9296283667 hartrees ##### • Virial Ratio = -<E>/<T> = 1.016536

  23. STO-3G SCF Results for Water One-Electron Energy Levels (Hartrees) • E( 1) = -20.611225487619 • E( 2) = -1.246335237157 • E( 3) = -0.634202849929 • E( 4) = -0.461580011790 • E( 5) = -0.393434714891 • E( 6) = 0.593547755409 • E( 7) = 0.699901990896 • Alpha-spin Orbitals for 5 Filled Orbitals by Column • # At-Orb 1 2 3 4 5 6 7 • 1 O 1s -0.995 -0.170 0.000 0.069 0.000 0.071 0.000 • 2 O 2s -0.038 0.851 0.000 -0.591 0.000 -0.927 -0.001 • 3 O 2px -0.003 0.095 0.484 0.458 -0.003 -0.464 0.762 • 4 O 2py -0.004 0.124 -0.371 0.597 -0.004 -0.603 -0.585 • 5 O 2pz 0.000 -0.001 0.000 -0.003 -1.000 0.004 0.000 • 6 H 1s 0.010 0.123 0.448 0.327 0.000 0.844 -0.806 • 7 H 1s 0.010 0.123 -0.449 0.327 0.000 0.843 0.807

  24. STO-3G Geometry Optimization • Retrieved previous Eigenvectors • The Two-Electron Integrals Have Been Computed • ##### Total Energy = -74.9296283891 hartrees ##### • Virial Ratio = -<E>/<T> = 1.016536 • Atom X(bohr) Y(bohr) Z(bohr) dE/dX dE/dY dE/dZ • 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 • 2 1.814137 0.000000 0.000000 -0.035671 0.000000 0.000000 • 3 -0.469539 1.752321 0.000000 -0.012809 -0.040369 0.000000 • Minimization No. 1 RMS Gradient = 0.055373 hartrees/bohr

  25. STO-3G Geometry Convergence • Retrieved previous Eigenvectors • The Two-Electron Integrals Have Been Computed • ##### Total Energy = -74.9343681281 hartrees ##### • Virial Ratio = -<E>/<T> = 1.017843 • Atom X(bohr) Y(bohr) Z(bohr) dE/dX dE/dY dE/dZ • 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 • 2 1.867011 0.000000 0.000000 -0.000002 0.000000 0.000000 • 3 -0.231435 1.852614 0.000000 -0.000093 -0.000014 0.000000 • Minimization No. 68 RMS Gradient = 0.000094 hartrees/bohr

  26. Water Structure • RASMOL Space-fill image

  27. STO-3G Geometry Convergence • Distance Matrix in Angstroms • O H H • O 0.0000 0.9880 0.9880 • H 0.9880 0.0000 1.4813 • H 0.9880 1.4813 0.0000 • A-B-C Arcs in Degrees for 3 Atoms • arc( 1, 2, 3)= 41.44 • arc( 1, 3, 2)= 41.44 • arc( 2, 1, 3)= 97.12 • arc( 2, 3, 1)= 41.44 • arc( 3, 1, 2)= 97.12 • arc( 3, 2, 1)= 41.44 • Retrieved previous Eigenvectors • The Two-Electron Integrals Have Been Computed • ##### Total Energy = -74.9343681369 hartrees ##### • Virial Ratio = -<E>/<T> = 1.017843 • GEOMETRY CONVERGED • CHEMSITE RESTART COORDINATES, COPY AND PASTE INTO INPUT FILE *.CTA • O 0.00000 0.00000 0.00000 • H 0.98798 0.00000 0.00000 • H -0.12242 0.98037 0.00000 • MOLUCAD RESTART COORDINATES, COPY AND PASTE INTO INPUT FILE *.XYZ • O 0.000000 0.000000 0.000000 • H 0.987981 0.000000 0.000000 • H -0.122421 0.980368 0.000000

  28. STO-3G Vibrations of Water • Approximate Energy Second-Derivative Matrix • 1 2 3 • 1 0.701E+00-0.117E-01-0.227E-01 • 2 -0.117E-01 0.107E+00-0.863E-01 • 3 -0.227E-01-0.863E-01 0.688E+00 • Mass-Weighted Force Constant Matrix • 1 2 3 • 1 0.696E+00-0.116E-01-0.225E-01 • 2 -0.116E-01 0.106E+00-0.856E-01 • 3 -0.225E-01-0.856E-01 0.682E+00 • Normal Modes Relative to Optimized Geometry • 1 2 3 • 1 0.719066 -0.694510 0.024500 • 2 0.083248 0.121085 0.989145 • 3 -0.689938 -0.709221 0.144885 • Normal Mode Eigenvalues (hartrees/(amu*bohr^2)) • 0.715875 0.674632 0.093624 • Harmonic Vibrational Frequencies and Corrections • J.A. Pople et. al. Int. J. Quantum Chem. S15, 269 (1981) • We apply 3-21G factor of 0.89 to any basis set initially. • We follow this with a Harmonic-Morse anharmonic analysis. • (K dyne/(amu*angstrom) ; frequency 1/cm ; 0.89 x 1/cm ) • ( 0.011146 ; 4349.34 ; 3870.92 ) • ( 0.010503 ; 4222.20 ; 3757.75 ) • ( 0.001458 ; 1572.89 ; 1399.87 )

  29. Harmonic-Morse Frequencies • Harmonic-Morse Anharmonic Analysis • See N.R. Zhang and D.D. Shillady, J. Chem. Phys. v100 p5230 (1994), Eq. A13. • (Three-point fit to Anharmonic Morse Potential) • Imaginary Frequencies will not be treated. • Summary of Harmonic and Anharmonic Vibrations • (K dyne/(amu*angstrom) ; frequency 1/cm ; 0.89x 1/cm ; Anharmonic/cm) • ( 0.011146 ; 4349.34 ; 3870.92 ; 3672.14 ) • ( 0.010503 ; 4222.20 ; 3757.75 ; 3584.42 ) • ( 0.001458 ; 1572.89 ; 1399.87 ; 1485.66 )

  30. STO-3G Charge Analysis • Dipole Moment Components in Debyes • Dx = 1.328207 Dy = 1.504385 Dz = 0.000000 • Resultant Dipole Moment in Debyes = 2.006815 • ******************************************************************************** • Computed Atom Charges • Q( 1)= -0.364 Q( 2)= 0.182 Q( 3)= 0.182 • ******************************************************************************** • Orbital Charges • 1.998545 1.934701 1.198554 1.232332 2.000000 0.817891 0.817977 • ******************************************************************************** • Mulliken Overlap Populations • # At-Orb 1 2 3 4 5 6 7 • 1 O 1s 2.046 -0.047 0.000 0.000 0.000 -0.001 -0.001 • 2 O 2s -0.047 2.181 0.000 0.000 0.000 -0.099 -0.100 • 3 O 2px 0.000 0.000 0.916 0.000 0.000 0.280 0.003 • 4 O 2py 0.000 0.000 0.000 0.956 0.000 0.000 0.277 • 5 O 2pz 0.000 0.000 0.000 0.000 2.000 0.000 0.000 • 6 H 1s -0.001 -0.099 0.280 0.000 0.000 0.677 -0.038 • 7 H 1s -0.001 -0.100 0.003 0.277 0.000 -0.038 0.677 • ******************************************************************************** • Total Overlap Populations by Atoms • O H H • O 8.0047 0.1797 0.1797 • H 0.1797 0.6767 -0.0385 • H 0.1797 -0.0385 0.6768 • ********* NORMAL FINISH OF PCLOBE ********* • (Use file lobe.xyz for RasMol or Viewerlite display.)

  31. 6-31G** Basis Set for Water • Nuclear Coordinates from Input • Atomic Core X Y Z Basis • 8. 0.319008 -0.256483 0.001576 3 • Z1s= -1.000 Z2s= 0.000 Z2p= -2.000 • Z3s= 0.000 Z3p= 0.000 Z3d= 2.490 • Z4sp= 0.000 Z4f= 0.000 • S-Symmetry Split Basis Set: 6 3 1 • P-Symmetry Split Basis Set: 3 1 • 1. 1.279004 -0.256602 -0.001146 3 • Z1s= -1.000 Z2s= 0.000 Z2p= -2.000 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • S-Symmetry Split Basis Set: 3 1 • P-Symmetry Split Basis Set: 1 • 1. 0.070645 0.670830 -0.001148 3 • Z1s= -1.000 Z2s= 0.000 Z2p= -2.000 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • S-Symmetry Split Basis Set: 3 1 • P-Symmetry Split Basis Set: 1 • Basis Size = 24 and Number of Spheres = 76 for 10 Electrons

  32. 6-31G** Energy for Water • Electronic Energy= -85.1877397903 a.u., Dif.= 0.0000000084 • Iteration No. 61 Energy Second Derivative =-0.00000000408301 • alpha = 0.900000 • Electronic Energy= -85.1877397929 a.u., Dif.= 0.0000000026 • Iteration No. 62 Energy Second Derivative =-0.00000000130947 • alpha = 0.900000 • Electronic Energy= -85.1877397938 a.u., Dif.= 0.0000000008 • Iteration No. 63 Energy Second Derivative =-0.00000000042013 • alpha = 0.900000 • Electronic Energy= -85.1877397941 a.u., Dif.= 0.0000000003 • Iteration No. 64 Energy Second Derivative =-0.00000000013486 • alpha = 0.900000 • Electronic Energy= -85.1877397942 a.u., Dif.= 0.0000000001 • Iteration No. 65 Energy Second Derivative =-0.00000000004297 • alpha = 0.900000 • Electronic Energy= -85.1877397942 a.u., Dif.= 0.0000000000 • ##### Total Energy = -76.0207191613 hartrees ##### • Virial Ratio = -<E>/<T> = 1.002842

  33. 6-31G** Geometry Convergence • ##### Total Energy = -76.0213106636 hartrees ##### • Virial Ratio = -<E>/<T> = 1.002001 • Atom X(bohr) Y(bohr) Z(bohr) dE/dX dE/dY dE/dZ • 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 • 2 1.782873 0.000000 0.000000 0.000049 0.000000 0.000000 • 3 -0.471593 1.719363 0.000000 0.000925 0.000283 0.000000 • Minimization No. 12 RMS Gradient = 0.000969 hartrees/bohr • ******************************************************************************** • Distance Matrix in Angstroms • O H H • O 0.0000 0.9434 0.9434 • H 0.9434 0.0000 1.5005 • H 0.9434 1.5005 0.0000 • A-B-C Arcs in Degrees for 3 Atoms • arc( 1, 2, 3)= 37.32 • arc( 1, 3, 2)= 37.32 • arc( 2, 1, 3)= 105.36 • arc( 2, 3, 1)= 37.32 • arc( 3, 1, 2)= 105.36 • arc( 3, 2, 1)= 37.32

  34. 6-31G** Charge Results • Dipole Moment Components in Debyes • Dx = 1.299590 Dy = 1.704617 Dz = 0.000000 • Resultant Dipole Moment in Debyes = 2.143514 • ******************************************************************************** • Computed Atom Charges • Q( 1)= -0.677 Q( 2)= 0.339 Q( 3)= 0.339 • ******************************************************************************** • Orbital Charges • 1.997645 0.913432 0.880216 0.871042 0.909996 1.159959 0.527623 0.580004 • 0.816037 0.003663 0.000775 0.001332 0.014237 0.001170 0.492203 0.127590 • 0.023338 0.007333 0.010959 0.492255 0.127604 0.008556 0.022093 0.010939 • ******************************************************************************** • Total Overlap Populations by Atoms • O H H • O 8.0457 0.3157 0.3157 • H 0.3157 0.3738 -0.0281 • H 0.3157 -0.0281 0.3739 • ********* NORMAL FINISH OF PCLOBE ********* • (Use file lobe.xyz for RasMol or Viewerlite display.)

  35. Correlated-SCF Method • Extending the basis set to more functions could lower the energy a little more but only to the Hartree Fock limit. A new way of computing the correlation energy is to replace the electron repulsion with a screened-repulsion. • Hartree Fock: (i,j/ (1/r12) /k,l) • Here: (i,j/ ((1 – exp(-a(r12)2)/r12) /k,l) • (i,j/ = Ψi(1)*Ψj(1), /k,l) = Ψk(2)* Ψl(2)

  36. Correlated-SCF 6-31G** Results • Using Chakravorty-Clementi-Otto-Type Correlation. • Radial cutoff = (1/r)*(exp(-omega*r*r)) • Shillady Formula: omega = 1.843076*sqrt(Z1*Z2*Z3*Z4) • Here Z1 is the effective scaling of each primitive sphere: • sc = sqrt(Ai/Ah), Z1 = sc*(1.0d0 + 0.400525*sc + 0.000000sc*sc) for all Z • Ah = 0.28294212105226 = Best H1s Gaussian Exponent. • The basis you have chosen (3) has a root-mean-square error of 0.020713 hartrees • compared to Quantum Monte Carlo energies for H2, LiH, BH, H2O and HF. • ******************************************************************************** • Retrieved previous Eigenvectors • The Two-Electron Integrals Have Been Computed • ##### Total Energy = -76.4249396609 hartrees ##### • Virial Ratio = -<E>/<T> = 1.005272 • Used Coulomb Hole Correlation with (A,b,c)=( 1.843076, 0.400525, 0.000000) • Atom X(bohr) Y(bohr) Z(bohr) dE/dX dE/dY dE/dZ • 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 • 2 1.760871 0.000000 0.000000 0.000027 0.000000 0.000000 • 3 -0.473364 1.696079 0.000000 0.000952 0.000294 0.000000 • Minimization No. 17 RMS Gradient = 0.000997 hartrees/bohr

  37. Correlated-SCF Optimized Geometry • Distance Matrix in Angstroms • O H H • O 0.0000 0.9318 0.9318 • H 0.9318 0.0000 1.4847 • H 0.9318 1.4847 0.0000 • A-B-C Arcs in Degrees for 3 Atoms • arc( 1, 2, 3)= 37.19 • arc( 1, 3, 2)= 37.19 • arc( 2, 1, 3)= 105.63 • arc( 2, 3, 1)= 37.19 • arc( 3, 1, 2)= 105.63 • arc( 3, 2, 1)= 37.19

  38. Correlated-SCF Charge Analysis • Dipole Moment Components in Debyes • Dx = 1.383061 Dy = 1.823089 Dz = 0.000000 • Resultant Dipole Moment in Debyes = 2.288343 • ******************************************************************************** • Computed Atom Charges • Q( 1)= -0.895 Q( 2)= 0.448 Q( 3)= 0.448 • ******************************************************************************** • Orbital Charges • 1.997382 0.891760 1.009190 0.889212 0.926495 1.160759 0.565952 0.610761 • 0.817485 0.005190 0.000839 0.001453 0.017184 0.001688 0.437031 0.082839 • 0.016333 0.006277 0.009742 0.437207 0.082909 0.007195 0.015395 0.009722

  39. Correlated-SCF Bonding • Total Overlap Populations by Atoms • O H H • O 8.3390 0.2782 0.2782 • H 0.2782 0.2953 -0.0213 • H 0.2782 -0.0213 0.2956

  40. Single-Excitation Excited States • Example of CIS excited states used for electric and magnetic transition moments to calculate oscillator strengths, natural rotational strengths and Faraday magnetically induced rotational strengths applied to formaldehyde, the prototypical ketone, to keep output short. • <i / μ/j> = magnetic transition, i→j <i/m/j> = electric transition, i→j

  41. Formaldehyde Structure • RASMOL Space-fill image

  42. Formadehyde STO-4G Input • Nuclear Coordinates from Input • Atomic Core X Y Z Basis • 6. 1.443750 0.000000 0.000000 1 • Z1s= 5.673 Z2s= 1.608 Z2p= 1.568 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • 8. 0.837973 -1.047436 -0.003630 1 • Z1s= 7.658 Z2s= 2.246 Z2p= 2.227 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • 1. 2.533744 -0.000815 -0.003265 1 • Z1s= 1.240 Z2s= 0.000 Z2p= 0.000 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • 1. 0.899456 0.944369 -0.003267 1 • Z1s= 1.240 Z2s= 0.000 Z2p= 0.000 • Z3s= 0.000 Z3p= 0.000 Z3d= 0.000 • Z4sp= 0.000 Z4f= 0.000 • Basis Size = 12 and Number of Spheres = 68 for 16 Electrons

  43. Optimized Geometry • Atom X(bohr) Y(bohr) Z(bohr) dE/dX dE/dY dE/dZ • 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 • 2 2.374870 0.000000 0.000000 0.000042 0.000000 0.000000 • 3 -1.132036 1.844326 0.000000 0.000601 0.000361 0.000000 • 4 -1.131963 -1.844415 0.000000 0.000597 -0.000358 0.000000 • Minimization No. 28 RMS Gradient = 0.000989 hartrees/bohr • ******************************************************************************** • Distance Matrix in Angstroms • C O H H • C 0.0000 1.2567 1.1452 1.1452 • O 1.2567 0.0000 2.0969 2.0969 • H 1.1452 2.0969 0.0000 1.9516 • H 1.1452 2.0969 1.9516 0.0000 • A-B-C Arcs in Degrees for 4 Atoms • arc( 3, 1, 2)= 121.56 arc( 3, 1, 4)= 116.88 arc( 3, 4, 1)= 31.56 • arc( 4, 1, 2)= 121.56 arc( 4, 1, 3)= 116.88 arc( 4, 3, 1)= 31.56 • Retrieved previous Eigenvectors • The Two-Electron Integrals Have Been Computed • ##### Total Energy = -113.2114299098 hartrees ##### • Virial Ratio = -<E>/<T> = 1.025594 • GEOMETRY CONVERGED

  44. STO-4G Formaldehyde Orbitals • Reference State Orbitals for 8 Filled Orbitals by Column • # At-Orb 1 2 3 4 5 6 7 8 • 1 C 1s -0.001 0.994 -0.107 0.187 0.000 -0.021 0.000 0.000 • 2 C 2s 0.008 0.027 0.228 -0.680 0.000 0.086 0.000 0.000 • 3 C 2px 0.006 0.000 0.127 0.207 0.000 0.498 0.000 0.000 • 4 C 2py 0.000 0.000 0.000 0.000 0.563 0.000 0.000 -0.246 • 5 C 2pz 0.000 0.000 0.000 0.000 0.000 0.000 -0.623 0.000 • 6 O 1s -0.995 0.000 -0.228 -0.108 0.000 0.098 0.000 0.000 • 7 O 2s -0.024 -0.005 0.801 0.475 0.000 -0.524 0.000 0.000 • 8 O 2px 0.005 0.000 -0.200 0.180 0.000 -0.662 0.000 0.000 • 9 O 2py 0.000 0.000 0.000 0.000 0.474 0.000 0.000 0.857 • 10 O 2pz 0.000 0.000 0.000 0.000 0.000 0.000 -0.661 0.000 • 11 H 1s -0.001 -0.006 0.025 -0.208 0.241 -0.143 0.000 -0.332 • 12 H 1s -0.001 -0.006 0.025 -0.208 -0.241 -0.143 0.000 0.332 • # At-Orb 9 10 11 12 • 1 C 1s 0.000 0.211 0.000 -0.078 • 2 C 2s 0.000 -1.445 0.000 0.565 • 3 C 2px 0.000 0.315 0.000 1.259 • 4 C 2py 0.000 0.000 -1.172 0.000 • 5 C 2pz -0.812 0.000 0.000 0.000 • 6 O 1s 0.000 -0.049 0.000 0.128 • 7 O 2s 0.000 0.322 0.000 -0.921 • 8 O 2px 0.000 -0.311 0.000 0.904 • 9 O 2py 0.000 0.000 0.318 0.000 • 10 O 2pz 0.781 0.000 0.000 0.000 • 11 H 1s 0.000 0.966 0.882 0.232 • 12 H 1s 0.000 0.966 -0.882 0.232

  45. CIS for Formaldehyde • ****************************************************************************** • Single-Excitation Configuration Interaction • J. A. Pople, Proc. Phys. Soc., A, Vol. LXVIII, p81, 1955 • ****************************************************************************** • Since this method is well known to predict transitions • to the blue of experimental electronic bands, we include • wavelengths scaled to fit the pi-pi* transition of benzene • at 262.53 nm calculated using the selected minimum basis. • This improves the pi-pi* excitations, but n-pi* and sigma • excitations are given better by the non-scaled values. • Transition type may be determined by noting non-zero • coefficients in the C.I. eigenvectors and the I->J orbital • list after assigning orbitals as n-, sigma- or pi-type in • the Reference Molecular Orbital matrix by columns. • For this calculation the pi* scaling factor is 0.6453

  46. Single-Excitations • CORE = 2 HOMO = 8 LUMO = 9 NBS = 12 • Excitation No. 1 = orbital ( 8 -> 9 ) • Excitation No. 2 = orbital ( 8 -> 10) • Excitation No. 3 = orbital ( 8 -> 11) • Excitation No. 4 = orbital ( 8 -> 12) • Excitation No. 5 = orbital ( 7 -> 9) • Excitation No. 6 = orbital ( 7 -> 10) • Excitation No. 7 = orbital ( 7 -> 11) • Excitation No. 8 = orbital ( 7 -> 12) • Excitation No. 9 = orbital ( 6 -> 9 ) • Excitation No. 10 = orbital ( 6 -> 10) • Excitation No. 11 = orbital ( 6 -> 11) • Excitation No. 12 = orbital ( 6 -> 12) • Excitation No. 13 = orbital ( 5 -> 9) • Excitation No. 14 = orbital ( 5 -> 10) • Excitation No. 15 = orbital ( 5 -> 11) • Excitation No. 16 = orbital ( 5 -> 12) • Excitation No. 17 = orbital ( 4 -> 9) • Excitation No. 18 = orbital ( 4 -> 10) • Excitation No. 19 = orbital ( 4 -> 11) • Excitation No. 20 = orbital ( 4 -> 12) • Excitation No. 21 = orbital ( 3 -> 9) • Excitation No. 22 = orbital ( 3 -> 10) • Excitation No. 23 = orbital ( 3 -> 11) • Excitation No. 24 = orbital ( 3 -> 12)

  47. The lowest 7 Excited States • SINGLET-C.I. Eigenvectors by Column • 1 2 3 4 5 6 7 • 1 -0.9743 0.0000 0.0000 0.2252 0.0000 0.0000 0.0000 • 2 0.0000 0.0000 0.0000 0.0000 0.5837 0.0000 0.0000 • 3 0.0000 0.0000 0.1496 0.0000 0.0002 0.0000 0.0000 • 4 0.0000 0.0000 0.0000 0.0000 -0.7727 0.0000 0.0000 • 5 0.0000 0.0000 0.9234 0.0000 0.0000 0.0000 0.0000 • 6 0.0000 0.0022 0.0000 0.0000 0.0000 -0.6030 0.1465 • 7 -0.0005 0.0000 0.0000 -0.0289 0.0000 -0.0002 0.0003 • 8 0.0000 0.0056 0.0000 0.0000 0.0000 0.5361 -0.6535 • 9 0.0000 -0.9934 0.0000 0.0000 0.0000 0.0443 0.0493 • 10 0.0000 0.0000 0.1035 0.0000 0.0000 0.0000 0.0000 • 11 0.0000 0.0000 0.0001 0.0000 0.0229 0.0000 0.0000 • 12 0.0000 0.0000 -0.3107 0.0000 0.0000 0.0000 0.0000 • 13 -0.2253 0.0000 0.0000 -0.9739 0.0000 0.0000 0.0000 • 14 0.0000 0.0000 0.0000 0.0000 0.1215 0.0000 0.0000 • 15 0.0000 0.0000 0.0935 0.0000 0.0001 0.0000 0.0000 • 16 0.0000 0.0000 0.0000 0.0000 -0.2161 0.0000 0.0000 • 17 0.0000 0.0726 0.0000 0.0000 0.0000 0.5891 0.7408 • 18 0.0000 0.0000 -0.0509 0.0000 0.0000 0.0000 0.0000 • 19 0.0000 0.0000 0.0000 0.0000 0.0081 0.0000 0.0000 • 20 0.0000 0.0000 0.0003 0.0000 0.0000 0.0000 0.0000 • 21 0.0000 -0.0889 0.0000 0.0000 0.0000 0.0047 0.0171 • 22 0.0000 0.0000 0.0307 0.0000 0.0000 0.0000 0.0000 • 23 0.0000 0.0000 0.0000 0.0000 0.0127 0.0000 0.0000 • 24 0.0000 0.0000 -0.0742 0.0000 0.0000 0.0000 0.0000

  48. Faraday A and B Terms • Buckingham-Stephens Equations • Cf. D. Shillady, UVa Ph.D. Thesis, 1970

  49. Faraday Band Shapes • Buckingham-Stephens Equations ( –B)

  50. Calculated Faraday Effect • NOTE: the signs of the MCD bands use the Schatz • Convention that the sign of the MCD of H2O is NEGATIVE at • 200 nm. Since Theta(H) = Consts.( A - B)H for H-field, • the SIGN OF THE B-Bands is MINUS (B/D) shown above; • D is always positive. • E= 3.450 ev, Lambda= 359.4 nm, (A/D)= 0.00000000, (B/D)= 0.00000000 • If (?) this is a pi->pi* band, the minimum basis result • can be fitted to the benzene singlet-(A1g->B2u) band and • predict Lambda= 557.0 nm • E= 8.447 ev, Lambda= 146.8 nm, (A/D)= 0.00000000, (B/D)=-0.00002093 • If (?) this is a pi->pi* band, the minimum basis result • can be fitted to the benzene singlet-(A1g->B2u) band and • predict Lambda= 227.5 nm • E= 10.983 ev, Lambda= 112.9 nm, (A/D)= 0.00000000, (B/D)= 0.00001073 • If (?) this is a pi->pi* band, the minimum basis result • can be fitted to the benzene singlet-(A1g->B2u) band and • predict Lambda= 174.9 nm • E= 12.054 ev, Lambda= 102.9 nm, (A/D)= 0.00000000, (B/D)= 0.00000000 • If (?) this is a pi->pi* band, the minimum basis result • can be fitted to the benzene singlet-(A1g->B2u) band and • predict Lambda= 159.4 nm

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