diatomics - COUNTS .e“ COUNTS e- COUNTS e... COUNTS e 0.0...

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Unformatted text preview: COUNTS .e“ COUNTS e- COUNTS e... COUNTS e 0.0 ELECTRON BINDING ENERGY (cV) 2MO 1.5 1.0 0.5 (3,0) ‘A (4.0) (2.0) (1v0)(5,0) (0,0) 0,5 1.0 1.5 2.0 CH PHOTOELECTRON KINETIC ENERGY (eV) 0. Electronic Structure of Diatomic Molecules Variation Theorem I Lijground stateHWQround stated I , , where H is J‘ wground stateWground statedT Given that < E >= ground state the full Hamiltonian, then the energy obtained using any wavefunction obeying the same boundary conditions as wgmund State will give an energy greater than or equal to <Eground state>, that IS (EgeneraP 2 {Eground state)- with the Variational Method, one varies an approximate wavefunction until the energy is minimized. For instance, suppose a wavefunction depends on the parameter 2’ (which might represent the effective nuclear charge seen by an electron due to screening by other electrons), then one finds the value of 2’ which gives the lowest energy by taking the derivative of the energy expression and setting it equal to zero. a J w*<z')l’-‘Iv(z')dr 6<E>_ iv‘(z')w(z')dr _0 62' _ aZ' _ Perturbation Theom Suppose, we have a problem to solve with a Hamiltonian (H) which is only slightly different than one for which we already know the answer. (Hm) In this case Perturbation Theory can be employed to find the solution. A H = Hm) + H' where H(O)w(0) : Emulw) Introduce a parameter (k) which takes the problem from the one you know how to solve to the desired problem upon going from 1:0 to i=1. A H=Hm+lH The final energies and wavefunctions can be represented as expansions in it _ (0) (1) (2) 2 __ (0) (1) (2) 2 E;_E‘+fixfiEnl+uawtm—wn+u%l+wnl+u Plugging these into the Schrodinger equations gives (fi=fim+AwaP+w91+w9firup4E9+E91+E9fiauxw9+w$x+w9fi+n) Collecting on terms of it gives finfi:gmwm=Emwm ammmlrgwm+gmwm=Emwm+Emww a useful tactic is to assume that the first-order perturbed part of the wavefuction is some (yet to be determined) linear combination of the range of zeroth-order solutions ms,” 2 ZaanEO). Plugging this equation for at,“ back into the first— J order Schrodinger equation gives “u (o " 0) m o 1 o 0 m o H W )+H( Zanng } = E( )w( ) +Ei )Zanlngi J' i which by the zeroth—order Schrodinger equation gives "u (0) m 0 (0) _ (1) 0) o) w (0) H w + Elam-E} )qu — E w‘ + E‘ Zeal-w; j j Now, multiply both sides by W" and integrate over all spatial coordinates. all of n the terms vanish except where n=j, so only one term survives from each sum above. n,n n j w wcoidHaHfiEgDaEQHa E‘O’ So the first-order perturbative correction to the energy is obtain by performing the above integral which involves only the zeroth—order wavefunction and the perturbative term of the Hamiltonian. For example the Helium atom Hamiltonian can be written in atomic units as 1‘ A 1 A. 1 1 A A 1 H=-——V21——V22————Z‘ —=H . 1 H . 2 — 2 ( ) 2 ( ) r1 r2 + r12 hydrogen-like( )+ hydrogen—like( )+ r1'2 So a perturbative treatment might take A A A A 1 0 1 H( ) = thdrogen-like + thdrogen—like and H( ) = r12 H2+ H; can be done exactly, see H. Wind, J. Chem. Phys. 42, 2371 (1965). One can profitany employ confocal elliptical coordinates to solve this problem. this coordinate system also makes it easy to analytically solve the overlap integral between orbitals on different atomic centers. Wind's equilibrium bond distance is 1.06 A, total energy is -O.6026342 Hartree, and bond energy is 0.1026342 Hartree or 2.7928 eV. OF CHEMICAL PHYSICS .[IE JOURNAL VOLUME 42,NUMDER7 [APRILIOGS Electron Energy for H2+ in the Ground State iI. WIND‘ MAC/LEA. Research. Gran/7, Czllham Laboratory, N1. Abingdon, Berkshire, England (Received 19 October 1964) The 155,, state of the hydrogen molecular ion is investigated. The result is given as a table in which the electronic energy for a two~Couloinb center is given in seven decimal places for values of internuclear Separation R up to 20 in steps of 0.05 an. HE field dissociation of Hz' is an important process for the injection of protons into mirror machines.‘" The probability of field dissociation depends on the energy of the uppermost vibrational levels. To calculate these energy levels it is necessary3 to have an accurate value for the electronic energy as a function of the internuclear separation R. As shown originally by Burrau,‘ the Schrodinger equation fora two-Coulomb center can be separated,and the electronic energy It thus calculath exactly. Bates 61 al.5 have calculated this energy for values of R up to 9 (atomic units are used throughout this paper), but this is not enough to determine the energies of the highest levels. Cohen 61 at,“ using a variational method, calculated the energy up to 16:20. However, this , result is not sufficiently accurate for our purpose. We ‘ have therefore calculated the electronic energy for the round state 150,, up to R240 in seven exact decimal ‘ places. The Schrodinger equation, expressed in confocal elliptic coordinates, )\ and a, and an azimuthal angle <1! is, in atomic units, 6 {(V—lfwi‘t a {(1 #E)a¢i+{ 1 + 1 Val: 0h 6) an a» V—l lip? 6d)- +liRzEtkz—MZ)+2RM¢=0- Writing #10» u, o) = AOQMWWW) and p2: —%R2E, the following equations are obtained: JED/dog: ~ m'-’<i>, (1) (l 2 dig J_ 2 _ m2 _ domain A+Pu2 ,_,.}M~0. (2) o d dAl , _mfi _ d)‘{(h2—1)d~)\I+{/i+ th—fiZW—Xl_ 1}A— 0, (3) _* On leave from POM. Instituul voor Plasma Fysica, lenhuizen, Jutphaas, Ncclerland. 'H. Postma, G. R. Haste, and J. L. Dunlap, Nucl. Fusion 3, 128 (1963). 2]. R. Hiskos, Phys. Rev. 122, 1207 (1961). “S. Cohen, J. R. l’liskcs, and R. J. Riddcll, Phys. Rev. 119, 1025 (1960). ‘ SD. Burrau, Kg]. Danskc, Vidcnskab. Selskab. Mat. Fys. Medd. 7»No.14(1927). “D. R. Bates, K. Ledsham, and A. L. Stewart, Phil. Trans. Roy. Soc. (London) A246, 215 (1954). ° 3. Cohen, D. L. Judd, and R. J. Riddcll, University of Cali- fornia, Lawrence Radiation Laboratory, Report No. CCRLSSOZ (June 1959). where A is the separation constant. We restrict our- selves to the case m= 0 since we calculate the electronic energy for the ground state 15:7". Equation (1) givesimmediately (Mo) =exp(im¢) = 1. To solve (2) we follow the method of Hylleraas,7 writing MOI) = :L‘zf’dfl) , P101) being Legendre functions. Vthn using the re- currence relation (1 —#2) Pn”(#) — zflPn/(fl)‘i’flf"'1+1)Pn(M):0: Eq. (2) gives a recurrence relation8 for 51: (l— 1)l (21—3) (21- 1) 2 [2 'firTJigifia 1') + (n+1) <21- 1) if?“ _ £2) (1+1) petal— »i ~z<l+u (2/+5)(2z+3 This recurrence relation is considered as an infinite set of homogeneous linear equations, in which the determinant of the coefficient matrix must be zero. This determinant contains p and A and, by equating it to zero, A is defined as a function of ,0. For the ground state l is even; for 1 odd we would find the solution for the 2pm, state. Since this determinant has an infinite number of terms, it is difficult to be certain of the number of terms which must be taken into account in order to achieve the required accuracy. However, only diagonal and neighboring terms exist and, in view of the simplicity and speed of calculating such a determinant by computer, we have considered a 50x50 determinant [notation D (50)]. Taking into account only D (10) does in fact lead to exactly the same values of A as a function of p in nine decimal places. Having calculated A as a function of 17, we substitute these values in (3) and now calculate E as a function of p or, since p2: —%RZE, E as a function of R. We Sign”: O. 7 E. A. liylleraas, Z. Physik '71, 739 (1931). “There appears to he a misprint in the original paper by Hylleraas. ln Er]. (9c) a factor C has been omitted in the last term on the left~hand side. 2371 2372 H. WIND TABLE I. The electronic energy E {Dr the hydrogen molzcular ion as a. function of the internuclear separation R. Atomic units are used.8 4.9919755 4.7275225 4.5000515 4.1555931. 4.9752121 5.10 4.7155555 10.70 4.5995599 15.10 4.5552725 4.9557215 5.15 4.7152110 10.15 4.5995519 15.15 4.5550111 4.9255252 5.20 4.7135953 10.75 4.5955515 11.20 4.5555155 4 5955570 5.25 4.7110195 10.25 4.195055: 15.25 4.5515110 4.5557059 5.10 4.705511. 10.15 4.1975555 15.10 4.5555011 4.5119027 5.35 4.7055125 10.15 4.5970715 15.15 4.5551905 4.5007559 5.50 4.7015555 10.1.0 4.5905751 4.5519755 4.7575525 5.1.5 4.7011529, 10.15 4.5951195 4.5557575 4.7199577 5.50 4.5990577 10.50 4.5955505 4.5555551 4.7025755 5.55 4.5955055 10.55 4.5911515 4.5555101 4.5719555 5.50 4.5955100 10.50 4.1957215 4.5511515 4.5509055 5.55 4.5925510 10.55 4.5952705 4.5519379 4.1111952 5.70 4.5955575 15.75 4.5915197 4.5517119 4.5523527 5.75 4.5152991 10.75 4.5915715 4.5555111 4.5555501 5.50 4.5152519 10.50 4.5929119 4.5515295 4.5271512 5.05 4.511.11117 10.55 4.5925955 4.2551275 4.5011515 5.90 4.5111791 10.95 4.5920515 4.5529105 4.5751555 5.95 4.5105579 10.95 4.5915129 4.2527125 4.1517553 5.00 4.5755157 11.00 4.5912555 4.5525155 4.5252151 5.05 4.5755217 11.05 4.5907579 4.5521512 4.5055027 5.10 4.5750559 11.10 4.5901715 4.5521572 4.3525335 5.12 4.5711055 11.15 4.5599590 4.551011: 4.3523075 5.20 4.5715991 11.20 4.5195505 4.5517525 4.3917951 5.25 4.5599255 11.25 4.5591529 4.5515725 4.1219715 5.15 4.5552915 11.30 4.5517950 4.5511512 4.1025050 5.15 4.5555570 11.15 4.5551579 4.5511951 4.2552592 5.50 4.1155111... 11.50 4.5579555 4.5510052 4.2151391 5.1.5 4.5515729 71.55 4.5575557 4.5505225 4.2559599 5.50 4.5520515 11.10 4.5571755 4.5555175 4.2521951 5.55 4.5555795 11.15 4.1557957 4.5505515 4.2159172 5.50 4.5591255 11.50 4.5155155 4.5502719 4.2001591 5.55 4.5577015 11.55 4.5550557 4.5500905 4.1559315 5.70 4.5555015 11.70 4.5155552 4.5579101. 4.1751555 5.75 4.5559125 11.75 4.5552991 4.5597115 4.1555092 5.50 4.5515571 11.50 4.5559152 4.5599112 4.1519072 5.55 4.5522559 11.15 4.5155705 4.5591752 4.1251215 5.90 4.5559712 11.90 4.5552112 4.5592002 4.1111115 5.95 4.5995995 11.95 4.5115159 4.5590251 4.1025112 7.00 4.5555111 12.00 4.5531015 4.5155515 4.0901021 7.05 4.5572255 12.05 4.5131575 4.5515755 . 4.0751251. 7.10 4.5550220 12.10 4.5525051 4.5115055 4.0155907 7.15 4.5550150 12.15 4.5125501 4.5515159 4.0553551 7.20 4.5515791 12.20 4.5521159 4 55-1551 4.0553955 7.25 4.5125157 72.25 .0 5117505 4.1579971 4.0557155 7.10 4.5515152 12.10 4.551....v 4.5575295 4.0251259 7.15 4.5551171 12.35 4.5511122 4.5575525 4.0112201 7.50 4.5192155 12.1.9 4.5507525 4.1575955 4.5011597 7.15 4.5351715 12.55 4.5505551 4.5575115 4.9915235 7.50 4.5172251 12.50 4.5101105 17.50 4.5171575 4.9555127 7.55 4.5150755 12.55 4.5795057 17.55 4.5570055 4.9755555 7.511 4.5150177 12.50 4.5795595 17.50 4.5555125 4.9555229 7.59 4.5110917 12.51 4.5791729 17.55 4.5555511 4.9550277 7.70 4.5131150 12.70 4.5755515 17.70 4.5555207 4.9590555 7.75 4.5121552 12.71 4.1755573 17.75 4.5551115 4.9515919 7.50 4.5312079 12.110 4.5752151 17.50 4.5152025 4.9115511 7.55 4.5102757 12.55 4.5779117 4.5150152 4.1255055 7.95 4.5295501 12.90 4.5775275 4.5515055 4.9152550 7.95 4.5255551 12.91 4.5771259 4.5517125 4.9155952 5.00 4.5275700 13.517 4.5770255 4.5155775 4.9557251 5.05 4.5255952 11.01 4.5757295 4.5555235 4.5957251 5.10 4.5255151 11.10 4.5755119 4.5552701 4.5595955 5.15 4.5759575 11.15 4.5751125 4.5511175 4.5512525 5.20 4.5751525 4.5715571. 4.5159550 4.575759. 5.25 4.5711100 71.25 4.5755515 4.511.515: 4.5705155 5.10 4.5225197 11.50 4.5752759 4.551.555: 4.5552125 5.15 4.5217212 11.11 4.5759955 4.5555152 4.555201? 5.90 4.5209155 11.50 4.5757151 4.5551579 4.5521152 5.55 4.5201557 11.1.5 4.5755199 4.5552205 4.5555675 5.50 4.5195952 11.50 4.5751575 4.5550717 4.1509517 5.15 4.5115555 11.51 4.5795525 4.5119277 4.5155571 5.50 4.5175975 11.50 4.5715595 4.5517525 4.5351521 5.55 4.5171555 11.55 4.5711159 4.1515151 4.5259225 5.70 4.5155522 11.75 4.5710700 4.9515957 4.5195255 5.75 4.5157291. 11.75 4.5725011 4.5211519 , 4.5155525 5.50 4.5150255 11.50 4.5725351 4.5512095 4.5099915 5.55 4.5155127 15.15 4.5722751 4.1110551 4.5052175 5.90 4.5115551. 11.90 4.5720150 4.5129210 4.5056125 5.72 4.5129710 11.95 4.5717555 4.5127552 4.7955559 9.00 4.5121055 15.00 4.5715975 4.5125592 4.7915523 9.05 4.5115557 111.55 4.5712112: 4.5121105 4.7571119 9.10 4.5159995 15.10 4.5709551. 4.1121712 4.7531211 9.15 4.5105555 15.11 4.1707155 4.5522153 4.7759975 9.20 4.5097255 15.20 4.5701555 4.5121002 4.7759555 9.25 4.5092005 15.25 4.5702151. 4.1519557 4.7715117 v.10 4.5555513 15.10 4.5599919 4.1515500 4.7571559 9.15 4.5071755 15.15 4.5597572 4.1115959 4.7515259 9.55 4.5572717 15.50 4.5595052 4.7515555 4.7597525 9.55 4.5555759 15.51 4.5592525 4.5115299 4.7551521 9.10 4.5050591. 15.90 4.5590212 4.5512919 4.7525525 9.55 4.5055055 15.55 4.5557552 4.5511555 4.7592251 9.50 4.505919: 15.55 4.5555559 4.5110159 4.7555721 9.55 4.5055553 15.55 4.5555152 4.1509057 4.7525957 9.75 4.5015050 15.717 4.551051: -5 5557755 4.7191927. 9.75 4.501251.) 15.75 4.5575597 4.1105579 4.7352015 9.50 4.5527059 15.10 4.5575195 4.1501199 4.7212005 9.15 4.5521515 15.55 4.5571915 4.1101925 4.7102075 9.90 4.5515105 15.90 4.5571555 4.5502555 4.7272515 9.95 4.5011015 15.95 4.1559195 4.5551197 4.7255201 4.5005757 4.5557155 4.1555153— " For camputer users, there is a FORTRAN deck 0[ the labia available on loan from Culham Laboratory. follow the method given by Jaffef‘ putting and taking A0) (H—ll'yffl—1)/(?\+1)]6_m. t w —1 y—Tég'm ' in which a: (R/IJ)-1. 'G.I.'Lfic,Z.Physik.87, 535 (1034). Then from (3) the following rccurrence relation i5 no“ » 46" l . l I , l I 0 D-OI 002 00! R 0-!“ 005 A RELATIVE ACCURACY OF APPROXIHATIOH FUR SHALL R ELECTRON ENERGY FOR H2+ 2373 'r, L l I 1 "° o lo 20 30 R 40 5'0 3 RELATIVE mom or mmxmmon FDR wire a FIG. 1. The results of the calculation are compared with results obtained from two approximations, (a) for small and (b) for large values of R. It can be seen that in both cases the approximations become exact. found : (i— l—o)2gi_1— {212+ (41)— 2a) l— A+p2- (2p+1)alg; We now consider this again as a coefficient matrix, the determinant of which must be zero. We have again taken D (50) into account although D (10) produces the same values of R. In either determinant, both that related to (2) and that related to (3), A and R, respectively, were adjusted until D (50) changed its Sign when A and R changed by less than 104’. Some 950 pairs of values for E and R were calculated. E was then calculated for values of R from 0.05 to 39.95 in steps of 0.05 by interpolation. To do so we took into account eight neighboring values of R for Which E had been calculated, using four on each side. Through these eight points a seventhpower poly« nominal was fitted, and this was then used to calculate E for the specified value of R. To check the accuracy interpolations using six, 10, and 12 points were also made for all the values of R. The dificrences between the values obtained using six, eight, 10, or 12 points +(l+l)figm=0- were in no case larger than 10”. The results for R up to 20 are presented in the Table I. For small values of R the system can be considered as a perturbed He‘r ion. BetheL0 has calculated that in this case the energy varies like ~2+§R1 We have calculated the energy for some 500 values of R<0.05. Figure 1(a) shows the accuracy of this approximation. For larger R the H2" ion consists essentially of 3. hydrogen atom in the ground state (energy —O.5) and a separated proton. Since the field produced by the proton is F: l/RZ, the energy due to the Stark shift of the ground state of the atom is — 9/4R‘. The energy of the two protons is +1/R, so that the electron energy for large R will approach —O_5—— (1/R)— (Q/4R‘). Figure 1(b) shows the accuracy of this approximation. In the table given by Cohen el 01.,5 the energy varies more like —0.5— (1/R)— (Z/R”) for large K. This may be due to the method of approximation used in their calculations. The author is grateful to Dr. J. R. Hiskcs and to Dr. A. C. Riviere for their suggestions and valuable discussions. “‘II. Bethe, “Quantenmechanik der Ein- und Zwei—Elektro- nenprobleme,” in Handth tier Physik, edited by H. Geiger and K. Schcel (Springer-Verlag, Berlin, 1933), Vol. 24/], p. 527. ,, H; (an b! dam. juggng Jee H. MM] ,7. (Ah-,JP/f/I. if. 237/ (my); 4% M, /0m/ e/bkzé‘m/ (om/m ,4; W 71 fl : WWW) MW! fig: 8&4? m:p/f;iZ_/h_ W72): % 9 {2(7) Mm {Mm jig " ’ WM/ " .. de aw: W733We7” W Ca 5;, P2: 5’67 X: 2f’7éti, 1'30 k {-151 r‘ l/ mom/r5 nae PO l H2 The class has made single-point, ab initio Cl calculations on the H2 molecule. The calculations are shown in the following figure. Only two points did not fall close to the fitted Morse potential curve. A Mathcad template follows which shows how to perform a nonlinear least squares fit of a Morse potential function to the data. Compare the class results to the literature values for harmonic vibrational frequency (4395.2 cm", see apprOpriate formula noting the missing square root symbol in Mortimer), equilibrium dissociation energy (4.476 eV), and equilibrium bond distance (0.7413 A). Not bad, heh? Ian-m (W) HF/cc-pVTZ M H-H Distance (angstroms) CHEM 533 Spring '97 the class‘ Cl caiculations on H2 input the data from chem533b.txt (distance-energy) (minus the two "bad" points) A :READPRN(chem533h2bm) npt37=rows(A) i::0..npts l rI. ’Aio Er. TA” fit to a Morse potential, eq. 142—36 in Mortimer DE=equitibrium bond dissooiafion energy a=curvature parameter=(k!(2*DE)}“.5 where k is Hooke's Law constant re=equilibrium bond distance con=constant to make dissociation limit zero in energy. i.e. the energy reference '12 F(r,DE,a,re,con) iTDE-[l T e'aUflMI ICOII L we want to minimize the sum of the squares of the deviations by varying the parameters SSE{DE.a,re,con) :: in! "E. — F= r_,DIE,z-i,re.ci)n“\2 L‘i I {1 I j i Minerr fitfing procedure initial guesses DE 1:.4 a *2 re '—.7 con :- 1.0 Given DE>0 re>Cl a>0 SSE(DE.a,re,con]=0 103] a (you need four constraints for four unknowns) 5 .'—- Minerr(DE,a,re, con) . . ['C | j results: the fitted parameters _ DE =0.168 Hartree a :22] 1/angstrorn rc =0-752 ang. con = el.l62 Hartree DIE-27.21 =4.56 eV ' am” 2032721450240” nih fran .. — -_ I 1'15- In14 1.1—. _- 1 nqfl_1n3 -...._-1 COD j results: the fitted parameters ’ DE=0.168 Harkee a=221 1Iangstrorn rc=0.752 ang. con=#l.l62Hartree DIS-27.21 =4.56 EV I a-m'“ 12052121450240“ .4 v 3 vib. freq- ‘r.:_.. - I , -__- v=l.476-10 Hz .. [B =4_922-10 cm“ - ' l-l "‘ - _ 2 a: ir__ _ “13606.10 27 2997910 A"; f 1‘" plot resultsrfil —.S,.Sl..3.l 0.9 '53“ -I E Li H Inc: 05 I 1.5 2 2.5 3 3.5 H-H (instance (angaromsJ # CI 6—318 SP hydrflgen O 1 H H l 0.96 Entering Link 1 = Ll.EXE PlD= 4470. Copyright (c) 1988,1990,1992,1993,1995, Gaussian, Inc. All Rights Reserved. This is part of the Gaussian 94(TM) system of programs. It is based on the the Gaussian 92(TM) system (copyright 1992 Gaussian, Inc.), the Gaussian 90(TM] system (copyright 1990 Gaussian, Inc.), the Gaussian 88(TM) system (copyright 1988 Gaussian, Inc.), the Gaussian 86(TM) system (copyright 1986 Carnegie Mellon University), and the Gaussian 82(TM) system (copyright 1983 Carnegie Mellon University). Gaussian is a federally registered trademark of Gaussian, Inc. This software is provided under written license and may be used, copied, transmitted, or stored only in accord with that written license. The following legend is applicable only to US Government contracts under DFARS: RESTRICTED RIGHTS LEGEND Use, duplication or disclosure by the US Government is subject to restrictions as set forth in subparagraph (c)(1)(ii) of the Rights in Technical Data and Computer Software clause at DFARS 252.227v7013. Gaussian, Inc. Carnegie Office Park, Building 6, Pittsburgh, PA 15l06 USA The following legend is applicable only to US Government contracts under FAR: RESTRICTED RIGHTS LEGEND Use, reproduction and disclosure by the US Government is subject to restrictions as set forth in subparagraph (c) of the Commercial Computer Software - Restricted Rights clause at FAR 52.227719. Gaussian, Inc. Carnegie Office Park, Building 6, Pittsburgh, PA 15106 USA Cite this work as: Gaussian 94, Revision 5.2, J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, G. Johnson, M. A” Robb, J. R. Cheeseman, T. Keith, A. Petersson, J. A. Montgomery, K. Raghavachari, A. AleLaham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, . S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. HeadeGordon, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Pittsburgh PA, 1995. LIFJOZCDCUK Jr*~k+1l—+++4r~kin):+~k++~kaki—+~Jc+=l~k~k+1k++~kir++~kir+~k+k++~kir+*~k* Gaussian 94: 486—Windows—G94ReVB.2 3—May—l995 25—Apr—1997 ****‘k******‘k*9r*‘k******‘k********************‘k*** Default route: MaxDisk=65536000 l/38=l/l; 2/12=2,l7=6,18=5/2; 3/5=1,6=6,11=9,25=1,30=l/1,2,3; 4//1; 5/5=2, 38:4/2; 8/623, 10:1,23=2,27=65536000/1,4; 9/27=65536000/13; 6/7=2, 8:2, 9:2, 10:2, 19:1/1; 99/5=l,9:1/99; Symbolic Z—matrix: Charge = 0 Multiplicity = l H H l l Z—MATRIX (ANGSTROMS AND DEGREES) CD Cent Atom N1 Length/X N2 Alpha/Y N3 Beta/Z J 1 1 H 2 2 H 1 1.000000( 1) Z—Matrix orientation: Center Atomic Coordinates (Angstroms) Number Number X Y Z l 1 0.000000 0.000000 0.000000 2 1 0.000000 0.000000 1.000000 Stoichiometry H2 Framework group D*H[C*(H.H)] Deg. of freedom 1 Full point group D*H NOp 8 Largest Abelian subgroup D2H NOp 8 Largest concise Abelian subgroup C2 NOp 2 Standard orientation: Center Atomic Coordinates (Angstroms) Number Number X Y Z l 1 0.000000 0.000000 0.500000 2 1 0.000000 0.000000 70.500000 Rotational constants (GHZ): 0.0000000 1002.9103340 1002.9103340 Isotopes: H—1,H—1 Standard basis: 6-31G (6D, 7F) There are 2 symmetry adapted basis functions of AG symmetry. There are 0 symmetry adapted basis functions of BlG symmetry. There are 0 symmetry adapted basis functions of BZG symmetry. There are There are There are symmetry adapted basis functions of 33G symmetry. symmetry adapted basis functions of AU symmetry. symmetry adapted basis functions of B1U symmetry. There are symmetry adapted basis functions of B2U symmetry. There are symmetry adapted basis functions of B3U symmetry. Crude estimate of integral set expansion from redundant integrals=l.000. Integral buffers will be 262144 words long. Raffenetti 1 integral format. Two—electron integral symmetry is turned on. OONOO 4 basis functions 8 primitive gaussians 1 alpha electrons 1 beta electrons nuclear repulsion energy 0.5291772490 Hartrees. Oneselectron integrals computed using PRISM. The smallest eigenvalue of the overlap matrix is l.486D-01 Projected INDO Guess. Initial guess orbital symmetries: Occupied (866) Virtual (SGU) (SGG) (SGU) Requested convergence on EMS density matrix=1.00D—O8 within 64 cycles. Requested convergence on MAX density matrix=1.00D—06. Keep R1 integrals in memory in canonical form, NReq= 418633. SCF Done: E(RHF) = -l.09480797231 A.U. after 5 cycles Convg = 0.1148D—1O -V/T = 2.1847 S**2 = 0.0000 Range of M.O.s used for correlation: l 4 NBasis= 4 NAE= l NBB= l NFC: O NFV= O NROrb= 4 NOAF l NOB: l NVfiF 3 NVB: 3 Spin components of T(2) and E(2): alpha—alpha T2 = 0.0000000000D+OO E2: 0.0000000000D+OO alpha—beta T2 = 0.1033869638D-Ol E2: LO.204O471822D-Dl beta—beta T2 = 0.0000000000D+00 E2: 0.0000000000D+OO ANorm= 0.1005156056D+Ol E2: —0.2040471822DeOl EUMP2= —O.11152126905300D+01 R2 and R3 integrals will be kept in memory, NReq= 400110. Iterations: 50 Convergence= 0.100D706 Iteration Nr. 1 ***************+****** DDlDir will call FoFMem 1 times, MxPair= 2 NAB= l NRA: O NBB: 0. The Euclidean norm of the A-vectors is 0.5266026D-Ol E3: —O.70763564D—02 3UMP3= —0.11222890470D+01 DE(CI)= —O.27l924l4D—Ol E(CI)= —O.11220003866D+Ol NORM(A)= 0.1012669OD+01 Iteration Nr. 2 *+******************** DDlDir will call FoFMem 1 times, MxPair= 2 NAB= l NAA= O NBB= U. The Euclidean norm of the Aevectors is 0.1114065D—01 DE(CI)= —O.31282520D—01 E(CI)= —O.11260904926D+Ol NORM(A)= 0.10144437D+01 Iteration Nr. 3 ****~k********9z*++***+* DDlDir will call FoFMem 1 times, MxPair= 2 NAB: l NAAF O NBB= O. The Euclidean norm of the Aavectors is 0.7910009DA03 DE(CI)= -O.31991025Dw01 E(CI): —O.11267989977D+Ol NORM(AJ= 0.10144343D+01 Iteration Nr. 4 ****~k***********‘k**‘k** DDlDir will call FoFMem 1 times, MxPair: 2 NABZ l NAA= 0 NBB= 0. The Euclidean norm of the Aevectors is 0.49109930~04 DE(CI)= —0.31973031D—01 E(CI)= —0.112678100330+01 NORM(A)= 0.1014427ZD+01 Iteration Nr. 5 *****+*****+*++*k+**** DDlDir will call FoFMem 1 times, MxPair= 2 NAB= l NAA= 0 NBB= 0. The Euclidean norm of the A—vectors is 0.1041326D—04 DE(CI)= -0.31970389D-01 E(CI)= -0.ll267783613D+01 NORM(AJ= 0.1014426BD+01 Iteration Nr. 6 ********************** DDlDir will call FoFMem 1 times, MxPair= 2 NAB: l NAA= 0 NEE: 0. The Euclidean norm of the A—vectors is 0.1646700D—05 DE(CI)= ~0.31970333D701 E(CI)= 70.11267783054D+01 NORM(A)= 0.10144269D+01 ***********i***i*+************i**********k*****************i*** Dominant configurations: *************+*******+* Spin Case I J A ABAB l l 2 Largest amplitude= 1.380—01 Value —0.l37885D+00 N03 **++*+**~k*++***+*+***+****~Jz++*+**+++*+*+*+++*+*+**+**+***+++++*++**+Jz* Population analysis using the SCF density. *************************+*****k*******************ik**********k****** Orbital Symmetries: Occupied (SGG) Virtual (SGU) (SGG) (SGU) The electronic state is lrSGG. Alpha occ. eigenvalues -- -O.52754 Alpha virt. eigenvalues -— 0.16771 0.90428 1.16227 Condensed to atoms (all electrons): 1 2 1 H 0.636185 0.363815 2 H 0.363815 0.636185 Total atomic charges: 1 1 H 0.000000 2 H 0.000000 Sum of Mulliken charges: 0.00000 Atomic charges with hydrogens summed into heavy atoms: 1 1 H 0.000000 2 H 0.000000 Sum of Mulliken charges: 0.00000 Electronic spatial extent (au): <R**2>= 6.6260 Charge: 0.0000 electrons Dipole moment (Debye): X: 0.0000 Y: 0.0000 Z: 0.0000 Tot: 0.0000 Quadrupole moment (Debye—Ang): XX: -2.4085 YY= —2.4085 ZZ= —1.6936 XY= 0.0000 XZ= 0.0000 YZ: 0.0000 Octapole moment (Debye—Ang**2): XXX: 0.0000 YYY= 0.0000 ZZZ= 0.0000 XYY= 0.0000 XXY= 0.0000 XXZ= 0.0000 X22: 0.0000 YZZ= 0.0000 YYZ= 0.0000 XYZ= 0.0000 Hexadecapole moment (Debye—Ang**3): XXXX= -2.4658 YYYY= -2.4658 ZZZZ= —4.5587 XXXY= 0.0000 XXXZ= 0.0000 YYYX= 0.0000 YYYZ: 0.0000 ZZZX= 0.0000 ZZZY= 0.0000 XXYY= 70.8219 XXZZ: —l.2042 YYZZ= —l.2042 XXYZ= 0.0000 YYXZ= 0.0000 ZZXY= 0 0000 N—N= 5.291772490000D-01 E—N=—3.ll7032851892D+00 KE= 9.241504380383D—01 Symmetry AG KE= 9.2415043803830-01 Symmetry BlG KE= 0.0000000000000+00 Symmetry BEG KE= 0.0000000000000+OO Symmetry 53G KE= 0.000000000000D+00 Symmetry AU KE= 0.000000000000D+00 Symmetry 310 KB: 3.626065746892D—l7 Symmetry B2U KE= 0.0000000000000+00 Symmetry B3U KB: 0.000000000000D+00 1|1|GINC—UNKISPIRCISD—FCI6—31GIH2|PCUSER125—Apr~1997|0ll# CI 6—316 SPI lhydrogenlI0,llHIH,l,l.l|Version=4B6-Windows—G94RevB.2|State=1—SGGIHF: e1.094808|MP2=—l.11521271MP3=—l.122289ICISD:—l.1267783[RMSD=1.148e—011 IPG=D*H [C*(H1.Hl)]||@ THE LARGE PRINT GIVETH, AND THE SMALL PRINT TAKETH AWAY. -— TOM WAITES Job cpu time: 0 days 0 hours 2 minutes 19.0 seconds. File lengths (MBytes): RWF= 29 Int: 0 02E: 0 Chk= l SCI: 1 Normal termination of Gaussian 94 The Covalent Bond Basic ideas in theoretical chemistry: Stability of molecules - calculate the energy of each atom individually, calculate the energy of the atoms arranged as the molecule (in practice this means find a minimum energy configuration), if the latter is lower in energy, then the molecule is stable, this energy difference is the atomic binding energy Born-Oppenheimer Approximation - the three body problem in quantum mechanics has escaped exact solution, we need simplifications: nuclei are much heavier than the electrons, their motions are slow compared to the motions of the electrons, it is assumed that the motions of the electrons are fast enough to adjust to any change of nuclei position, therefore nuclear motions can be separated from the electronic problem. The electronic problem is solved as a function of nuclear positions. Variation theorem - the energy obtained with an approximate wavefunction (with the same boundary conditions, full Hamiltonian) will be greater than the energy obtained from the true wavefunction. This serves as the basis for finding the structure of molecules by minimizing the energy. ...
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