Quantum course

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Quantum course By one of the Famous Professors in Chemistry department , Faculty of Science , Tanta , University

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Quantum course

  1. 1. By: Prof .Dr Mohamed Khaled
  2. 2. Quantum Chemistry Quantum Chemistry is the application of quantum mechanics to solve problems in chemistry. It has been applied in different branches of chemistry Physical Chemistry: To calculate thermodynamic properties, interpretation of molecular spectra and molecular properties (e.g. Bond length, bond angles, …..etc.). Organic Chemistry: To estimate the relative stabilities of molecules, and reaction mechanism. Analytical Chemistry: To Interpret of the frequency and intensity of line spectra. Inorganic Chemistry: To predict and explain of the properties of transition metal complexes.
  3. 3. Physical Chemistry Inorganic Chemistry Organic Chemistry Photochemistry Polymer Surface and Catalysis Drug Design Toxicity
  4. 4. Historical background of quantum mechanics : Nature of light: Hertz, 1888, has showed that light is electromagnetic waves. λ=c/ν where, λ is the wavelength, c is the speed of light =2.998 x 1010 cm/sec, ν is the frequency cm/sec. Max Plank has assumed only certain quantities of light energy (E) could be emitted. E = hν Where, h is Plank’s constant = 6.6 x 10-27 erg.sec. The energy is quantized. Photoelectric effect : Light comes out by shining surface in vacuum. In 1905, Einstein, light can exhibit particle like behavior, called photons. Ephoton = hν hν = W + ½ mv2 where, W is the work function (minimum energy required to take electron out). ½ mv2 is the kinetic energy of emitted electron. From above, it is assumed that the {Light looks like a particle and a wave}
  5. 5. Nature of Matter: Rutherford and Geiger have found that some α-particles bounced right back from golden foil, have small positive nucleus in atom. In 1913, Bohr has studied the H-atom and assumed that the energy of electron is quantized, ν = ∆E / h where, ν is the frequency of absorbed or emitted light, ∆E is the energy difference between two states. In 1923, DeBroglie has suggested that the motion of electrons might have a wave aspect. λ = h / mv = h / p where, m is mass of electron. p is a particle momentum. Accordingly, it has been suggested that electrons behave in some respect like particles and in some others like waves. This is what is called a Particle –Wave Duality. The question arises, how can an electron be a particle, which is a localized entity, and a wave, which is nonlocalized? The answer is No, neither a wave nor a particle but it is something else. The Classical physics has failed to describe the microscopic particles.
  6. 6. The question arises, how can an electron be a particle, which is a localized entity and a wave which is a non-localized? The answer is No, neither a wave nor a particle but it is something else.
  7. 7. Heisenberg Uncertainty Principle: "It is impossible to determine precisely and simultaneously the momentum and the position of an electron The statistical definition for the uncertainties is: ∆x . ∆px ≥ ħ / 2 where, = ħ / 2π ∆x . ∆px ≥ h / 4π Werner Heisenberg Nobel prize 1932
  8. 8. Wave Function: To describe the state in quantum mechanics, we postulate the existence of a function of the coordinates called the wave function (State function), ψ. ψ = ψ (x, t) It contains all information about a system. The probability of finding a particle in a given place can be given by ψ (Probability description).
  9. 9. What ψ (x) means? - ψ is an amplitude, sometimes complex function, not measurable, imaginary value. - ψψ* is a complex function, which may be real, and positive. - ψ has no physical meaning but ψψ* is the probability of locating the electron at a given position. If the probability of a certainty is defined as unity, this means: If we have two different wave functions, ψ1 and ψ2 will be Normalized function when: and But if or The function is called orthogonal function. But if Where, (called Kronecker Delta) is equal zero when i ≠ j and equal one when i = j, the function is called orthonormalized function.
  10. 10. Time-dependent Schrödinger equation: we postulate the existence of a function of the coordinates called the wave function (State function), ψ. For one particle, one-dimensional system: ψ= ψ (x, t) It contains all information about a system. The probability of finding a particle in a given place can be given by ψ (Probability description). Born postulates | ψ (x,t) |2 dx is the probability of finding a particle at position x and at time t ( Probability density ). ψ must satisfy Schrödinger equation. As t passes, ψ changes to differential equation: −  ∂Ψ ( x, t ) −  2 ∂ 2 Ψ ( x, t ) = + V ( x , t ) Ψ ( x, t ) 2 i ∂t 2m ∂x where, i= , m = particle mass, V(x,t) = potential energy. This is called Time-dependent Schrödinger equation. Erwin Schrödinger Nobel prize 1933
  11. 11. Schrödinger equation can be solved by the technique called separation of variables: the partial derivatives of this equation: Making the substitution in equation 2: Dividing by
  12. 12. Taking the left side of equation (3): On integration: ln C is a constant of integration
  13. 13. One of the properties of the wave function, it is a complex, i.e. where, is a complex conjugate of The complex conjugate of a function is the same function with a different sign of imaginary value. for stationary state is called the Probability Density ( Time-independent wave function).
  14. 14. By equating the right side of equation (3) to a constant E, we have: Time-independent Schrödinger equation for a single particle of mass m moving in one dimension. The constant E has the dimension of energy. In fact, it is postulated that E is the energy of the system.
  15. 15. Operators: Basis of quantum mechanics set up around two things: 1- Wave function, which contains all information about the system. 2- Operators which are rules whereby given some function, we can find another. This operator is called the Hamiltonian operator for the system. Kinetic energy = Ĥ = Kinetic energy + Potential energy = So, the Eigen value equation: Ĥ ψ(x) = E ψ(x)
  16. 16. -A particle in one-dimensional box: I V=0 III II X =0 Ψ=0 X =1 X −  2 ∂ 2ψ ( x) + ∞ψ ( x) = Eψ ( x) 2 2m ∂x ∂ 2ψ ( x) 2m + 2 ( E − ∞)ψ ( x) = 0 ∂x 2  We conclude that ψ(x) is zero outside the box: ψ I(x) = zero ψ III(x) = zero
  17. 17. For region II (inside the box), x between zero and l, the potential energy V(x) is zero, and the Schrödinger equation becomes n= 1,2,3,………
  18. 18. Fig. 3.1. The wave functions for the 0ne-dimensional particle-ina-box Fig. 3.2. The probability densities in 0nedimensional particle-in-a-box
  19. 19. II- The Harmonic Oscillator: 1- Try to understanding of molecular vibrations, their spectra and their influence on thermodynamic properties. 2- Providing a good demonstration of mathematical techniques that are important in quantum chemistry. V E Velocity=0 -a 0 a x Fig. 4 The Parabolic Potential Energy of the Harmonic Oscillator. The classically allowed (|x| ≤ a) and forbidden (|x| > a) regions for the Harmonic Oscillator
  20. 20. The classical force F is: F= -kx Where, F is a restoring force, k is a force constant, and x is a displacement on x-axis. F= By integration: Where, = - kx V(x) = ½ kx2
  21. 21. The Schrödinger equation ψ(x) = E ψ(x), after multiplication on by 1  α  4 −αx 2 2 ψ0 =   ∈ π  The energy of a harmonic oscillator is quantized. En = (n + ½ ) h ν where n= 0,1,2…
  22. 22. III- The Hydrogen atom: Ignoring interatomic or intermolecular interactions, The isolated hydrogen atom is a two-particle system. Instead of treating just the hydrogen atom, we consider a slight more general problem, the hydrogen-like atom. An exact solution of the Schrödinger equation for atoms with more than one electron cannot be obtained because of the interelectronic repulsions. V = -Z é 2 / r Where, V is the potential energy, Zé is the charge of nucleus, (For Z=1, we have the hydrogen atom, for Z=2 the He + ion, for Z=3, the Li+ ion, etc…). é is the proton charge in statocoulombs or as: é≡ where, e is the proton charge in coulomb. To deal with the internal motion of the system, we introduce µ as the mass of the particle. µ = m e mn / m e + m n where, me and mn are the electronic and nuclear masses.
  23. 23. where, is Laplacian operator: So, the time-independent Schrödinger equation is: z me z = r cos θ x = r sin θ . cos φ y = r sin θ . sin φ mn x θ r φ y
  24. 24. To solve this equation, we have to know that this wave is a spherical one, so, we should convert the Cartesian coordinates to spherical polar coordinates. There are two different variables in Schrödinger equation, one is the radial variable (r) and the other is the angular variable . This is called Bohr radius. According to the Bohr theory, it is the radius of the circle in which the electron moved in the ground state of the hydrogen atom.
  25. 25. The wave function for the ground state of the H-atom, where n=1, l=0, and m=0 The bound-state energy levels of the hydrogen-like atom are given by Substituting the values of the physical constants into the energy equation of H-atom, we find for (Z=1) ground state energy: E = -13.598 eV (eV= electron volt)
  26. 26. Shapes of electron cloud: Probability densities for some hydrogen-atom states
  27. 27. The overlap integral between two wave functions can be represented as S ij Sij = ∫ψi ψj dτ Three different kinds of overlap are shown in Fig. ( 9). Positive (Bonding) Fig. 9 Three different kinds of overlap between two wave functions, ψ i and ψ j
  28. 28. σ and π bonds
  29. 29. Molecular Orbital Theory : The MO Theory has five basic rules: 1-The number of molecular orbitals = the number of atomic orbitals combined of the two MO's, one is a bonding orbital (lower energy) and one is an anti-bonding orbital (higher energy) 2-Electrons enter the lowest orbital available 3-The maximum # of electrons in an orbital is 2 (Pauli Exclusion Principle) 4- Electrons spread out before pairing up (Hund's Rule)
  30. 30. Heteronuclear molecules: Hydrogen Fluoride: Z F Y Table 5 1σ 2σ 3π 4π 5σ 2s -0.93 0.47 0 0 0.55 2px -0.009 -0.68 0 0 0.80 2py 0 0 1.0 0 0 2pz 0 0 0 1.0 0 1sH -0.16 -0.57 0 0 -1.05 E(eV) -40.17 -15.39 -12.64 -12.64 3.20 H X
  31. 31. 2 0 0 0 NN 0.000000 .000000 .000000 2 1.9237-20.330 2 1.9170-14.540 0 0.0000 00.000 .0000 .00000.0000 7 30 2 0 3 1.020000 0.000000 0.000000 2 1.9237-20.330 2 1.9170-14.540 0 0.0000 00.000 .0000 .00000.0000 7 30 2 0 3 0 2 1 .010000 3.000000
  32. 32. Quantum chemical studies of the correlation diagram of N2 molecule with standard parameters: __________________________________________________________________________________ ATOM X Y Z S P D CONTRACTED D N EXP COUL N EXP COUL N EXPD1 COUL C1 C2 EXPD2 AT EN S P D N 1 .00000 .00000 .00000 2 1.9237 -20.3300 2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 3 N 2 1.20000 .00000 .00000 2 1.9237 -20.3300 2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 3 0CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON= 2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC=.00000 0DISTANCE MATRIX 1 2 1 .0000 1.200 2 1.200 .0000 0TWO BODY REPULSION ENERGY MATRIX 1 2 1 .0000 4.4351 2 4.4351 .0000 4.43514361 0SPIN= 0
  33. 33. ENERGY LEVELS (EV) E( 1) = 15.24199 0 E( 2) = -7.40075 0 E( 3) = -7.40075 0 E( 4) = -14.54000 2 E( 5) = -14.54000 2 E( 6) = -16.54000 2 E( 7) = -18.54000 2 E( 8) = -27.51826 2 0 ENERGY= -187.27312236 EV. 5 ORBITALS FILLED 0 HALF FILLED 0WAVE FUNCTIONS 0MO'S IN COLUMNS, AO'S IN ROWS 1 2 3 4 5 6 7 8 1 2 3 .0000 .0000 .0000 -1.2989 .0000 .0000 .0000 -.7443 .0000 .0000 .0000 .7443 .0000 .0000 .0000 -1.2989 .0000 .0000 .0000 .7443 .0000 .0000 .0000 -.7443 4 .0000 .0000 .0000 .5234 .0000 .0000 .0000 .5234 5 .0000 .0000 .5234 .0000 .0000 .0000 .5234 .0000 6 7 8 .0000 .5068 .5223 .5144 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 -.5068 .5223 -.5144 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
  34. 34. 0REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM 1 2 1 4.0437 2.7000 2 2.7000 4.0437 0REDUCED CHARGE MATRIX, MO'S IN COLUMNS, ATOMS IN ROWS 1 2 3 4 5 6 7 8 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0ATOM N 1 N 2 NET CHG. .00000 .00000 S ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION X Y Z X2-Y2 Z2 XY XZ YZ 1.40947 1.59053 1.00000 1.00000 1.40947 1.59053 1.00000 1.00000
  35. 35. Quantum chemical studies of the correlation diagram of N2 molecule with the hybridized parameters: __________________________________________________________________________________ ATOM X Y Z S N EXP COUL N 1 .00000 .00000 .00000 2 1.9237 -20.3300 N 2 1.02000 .00000 .00000 2 1.9237 -20.3300 0CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON= 0DISTANCE MATRIX 1 2 1 .0000 1.0200 2 1.0200 .0000 0TWO BODY REPULSION ENERGY MATRIX 1 2 1 .0000 4.4351 2 4.4351 .0000 4.43514361 0SPIN= 0 P D CONTRACTED D N EXP COUL N EXPD1 COUL C1 C2 EXPD2 AT EN S P D 2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 3 2 1.9170 -14.5400 0 .0000 .0000 .00000 .00000 .0000 7 30 2 0 3 2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC=.00000
  36. 36. ENERGY LEVELS (EV) E( 1) = 115.24199 0 E( 2) = -7.40075 0 E( 3) = -7.40075 0 E( 4) = -14.07184 2 E( 5) = -18.08657 2 E( 6) = -18.08873 2 E( 7) = -18.08873 2 E( 8) = -27.51826 2 0 ENERGY= -187.27312236 EV. 5 ORBITALS FILLED 0 HALF FILLED 0WAVE FUNCTIONS 0MO'S IN COLUMNS, AO'S IN ROWS 1 2 3 4 5 6 7 8 1 2 3 -1.6157 .0000 .0000 -1.2989 .0000 .0000 .0000 -.7443 -.4460 .0000 -.4460 .7443 1.6157 .0000 .0000 -1.2989 .0000 .0000 .0000 .7443 -.4460 .0000 .4460 -.7443 4 5 .3190 -.4437 -.6477 .5144 .0000 .0000 .0000 .0000 .3190 .4437 .6477 -.5144 .0000 .0000 .0000 .0000 6 .0000 .0000 .3426 .5068 .0000 .0000 .3426 .5068 7 8 .0000 .5223 .0000 .1333 .5068 .0000 -.3426 .0000 .0000 .5223 .0000 -.1333 .5068 .0000 -.3426 .0000
  37. 37. REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM 1 2 1 4.0437 1.9126 2 1.9126 4.0437 0REDUCED CHARGE MATRIX, MO'S IN COLUMNS, ATOMS IN ROWS 1 2 3 4 5 6 7 8 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0ATOM N 1 N 2 NET CHG. .00000 .00000 S ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION X Y Z X2-Y2 Z2 XY XZ YZ 1.40947 1.59053 1.00000 1.00000 1.40947 1.59053 1.00000 1.00000
  38. 38. 3 0 0 0 OHH 2 2.2459-28.480 2 2.2266-13.620 0 0.0000 0.000 0.00000.00000.0000 8 35 2 4 0 1 1.2000-13.600 0 0.0000 0.000 0 0.0000 0.000 0.00000.00000.0000 1 10 1 0 0 1 1.2000-13.600 0 0.0000 0.000 0 0.0000 0.000 0.00000.00000.0000 1 10 1 0 0 0.00000000 0.00000000 1 2 0.99000 52.25000 1 3 0.99000 52.25000 0 1 2 2 1.000 132.0000 0.00000000 0.00000 180.00000
  39. 39. Quantum chemical studies of the electronic structure of H2O molecule: __________________________________________________________________________________ IEXIT = 0 ETA .00000 ETA 180.00000 ATOM X Y Z S P D CONTRACTED D N EXP COUL N EXP COUL N EXPD1 COUL C1 C2 EXPD2 AT EN S P D O 1 .00000 .00000 .00000 2 2.2459 -28.4800 2 2.2266 -13.6200 0 .0000 .0000 .00000 .00000 .0000 8 35 2 4 0 H 2 .78278 .00000 -.60610 1 1.2000 -13.6000 0 .0000 .0000 0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 0 H 3 -.78278 .00000 -.60610 1 1.2000 -13.6000 0 .0000 .0000 0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 0 0CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON= 2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC=.0000 0DISTANCE MATRIX 1 2 3 1 2 3 .0000 .9900 .9900 .9900 .0000 1.5656 .9900 1.5656 .0000 0TWO BODY REPULSION ENERGY MATRIX 1 2 3 1 .0000 1.0063 1.0063 2 1.0063 .0000 .0345 3 1.0063 .0345 .0000 2.04704595 0SPIN= 0
  40. 40. ENERGY LEVELS (EV) E( 1) = 14.73419 0 E( 2) = 3.38040 0 E( 3) = -13.62000 2 E( 4) = -14.43845 2 E( 5) = -16.97488 2 E( 6) = -31.03678 2 ENERGY= -150.09316161 EV. 0 4 ORBITALS FILLED 0 HALF FILLED 0WAVE FUNCTIONS 0MO'S IN COLUMNS, AO'S IN ROWS 1 2 3 4 5 6 1 -.9712 .0000 .0000 .5517 .8029 .8029 2 .0000 .9247 .0000 .0000 -.8588 .8588 3 4 5 .0000 .2319 .0000 .0000 .0000 -.6834 1.0000 .0000 .0000 .0000 .9213 .0000 .0000 -.1291 -.3815 .0000 -.1291 .3815 6 .7793 .0000 .0000 .0224 .2029 .2029
  41. 41. REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM 1 2 3 1 5.9549 .6598 .6598 2 .6598 .4068 -.0882 3 .6598 -.0882 .4068 0REDUCED CHARGE MATRIX, MO'S IN COLUMNS, ATOMS IN 1 2 3 1 2 3 4 5 6 .6344 .7508 2.0000 1.8608 1.2492 1.5048 .6828 .6246 .0000 .0696 .3754 .2476 .6828 .6246 .0000 .0696 .3754 .2476 0ATOM O 1 H 2 H 3 ROWS NET CHG. -.61478 .30739 .30739 ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION S X Y Z X2-Y2 Z2 XY XZ YZ 1.56003 1.24916 2.00000 1.80558 .69261 .69261
  42. 42. 4 0 0 0 NHHH 2 1.6237-17.830 2 1.6170-12.040 0 0.0000 0.000 0.00000.00000.0000 7 30 2 3 0 1 1.2000-13.600 0 0.0000 0.000 0 0.0000 0.000 0.00000.00000.0000 1 10 1 0 0 1 1.2000-13.600 0 0.0000 0.000 0 0.0000 0.000 0.00000.00000.0000 1 10 1 0 0 1 1.2000-13.600 0 0.0000 0.000 0 0.0000 0.000 0.00000.00000.0000 1 10 1 0 0 0.00000000 0.00000000 1 2 1.22000 60.00000 1 3 1.22000 180.00000 1 4 1.22000 60.00000 0 1 2 1 0.010 1.5000 0.00000000 0.00000 90.00000 180.00000
  43. 43. ATOM D N H H H 1 2 3 4 X .00000 1.10000 .00000 -1.10000 Y .00000 .00000 .00000 .00000 Z .00000 -.60000 1.20000 -.60000 N EXP S COUL N EXP P COUL D CONTRACTED D N EXPD1 COUL C1 C2 EXPD2 AT EN S P 2 1.6237 -17.8300 2 1.6170 -12.0000 0 1 1.2000 -13.6000 0 .0000 .0000 0 1 1.2000 -13.6000 0 .0000 .0000 0 1 1.2000 -13.6000 0 .0000 .0000 0 0DISTANCE MATRIX 1 2 3 4 1 .0000 1.2200 1.2200 1.2200 2 1.2200 .0000 2.1131 2.1131 3 1.2200 2.1131 .0000 2.1131 4 1.2200 2.1131 2.1131 .0000 .0000 .0000 .0000 .0000 .0000 .00000 .00000 .00007 30 2 3 0 .0000 .00000 .00000 .0000 1 10 1 0 0 .0000 .00000 .00000 .0000 1 10 1 0 0 .0000 .00000 .00000 .0000 1 10 1 0 0
  44. 44. ENERGY LEVELS (EV) E( 1) = 16.00000 0 E( 2) = 5.00000 0 E( 3) = 5.00000 0 E( 4) = -12.00000 2 E( 5) = -17.00000 2 E( 6) = -17.00000 2 E( 7) = -23.00000 2 ENERGY= -131.90074095 EV. 0WAVE FUNCTIONS 0MO'S IN COLUMNS, AO'S IN ROWS 1 2 3 4 5 6 7 1 -1.2702 .0000 .0000 .0000 .0000 .0000 .6009 2 .0000 .0000 1.0340 .0000 -.5499 .0000 .0000 3 .0000 .0000 .0000 -1.0000 .0000 .0000 .0000 4 .0000 1.0340 .0000 .0000 .0000 .5499 .0000 5 .6891 .1677 -.9271 .0000 -.4096 -.2804 .2515 6 .6891 -.8867 .0000 .0000 .0000 .4949 .2515 7 .6891 .7190 .6088 .0000 .4476 -.2145 .2515 0REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM 1 2 3 4 1 3.9387 .7613 .7613 .7612 2 .7613 .6193 -.0267 -.0267 3 .7613 -.0267 .6192 -.0267 4 .7612 -.0267 -.0267 .6192 0ATOM NET CHG. ATOMIC ORBITAL OCCUPATION FOR GIVEN MO OCCUPATION S X Y Z X2-Y2 Z2 XY XZ YZ N 1 -.08064 1.12890 .97589 2.00000 .97586 H 2 .02683 .97317 H 3 .02689 .97311 H 4 .02693 .97307
  45. 45. ATOM X Y Z S D CONTRACTED D N EXP COUL N EXP COUL N EXPD1 COUL C1 C2 EXPD2 AT EN S P D F 1 .00000 .00000 .00000 2 2.5630 -37.5800 2 2.5500 -17.4200 0 .0000 .0000 .00000 .00000 .0000 9 40 2 5 0 H 2 .00000 .00000 -1.20000 1 1.2000 -13.6000 0 .0000 .0000 0 .0000 .0000 .00000 .00000 .0000 1 10 1 0 0 0CHARGE= 0 IEXIT= 0 ITYPE= 0 IPUNCH= 0 CON= 2.250 AU= .52920 SENSE= 2.000 MAXCYC= 25 DELTAC= .00000100 0DISTANCE MATRIX 1 2 1 .0000 1.2000 2 1.2000 .0000 0TWO BODY REPULSION ENERGY MATRIX 1 2 1 .0000 .0846 2 .0846 .0000 .08463978 0SPIN= 0 0 ENERGY LEVELS (EV) E( 1) = -5.49648 0 E( 2) = -17.42000 2 E( 3) = -17.42000 2 E( 4) = -18.31403 2 E( 5) = -38.02579 2 ENERGY= -182.27500191 EV. 4 ORBITALS FILLED 0 HALF FILLED 0WAVE FUNCTIONS 0MO'S IN COLUMNS, AO'S IN ROWS 0 1 2 3 4 5 1 -.4135 .0000 .0000 -.1327 .9547 2 .0000 .0000 -.7071 .0000 .0000 3 .0000 -.7071 .0000 .0000 .0000 4 .5194 .0000 .0000 -.8947 .0126 5 1.0364 .0000 .0000 .2825 .1295 P
  46. 46. 0REDUCED OVERLAP POPULATION MATRIX, ATOM BY ATOM 1 2 1 7.4596 .3472 2 .3472 .1931 0REDUCED CHARGE MATRIX, MO'S IN COLUMNS, ATOMS IN 1 2 3 4 5 1 .3667 2.0000 2.0000 1.7383 1.8949 2 1.6333 .0000 .0000 .2617 .1051 0ATOM NET CHG. OCCUPATION F 1 H 2 -.63325 .63325 ATOMIC ORBITAL OCCUPATION ROWS FOR GIVEN MO S X Y Z X2-Y2 1.90871 2.00000 2.00000 1.72454 .36675 Z2 XY XZ YZ

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