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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 firstorder perturbed part of the wavefuction
is some (yet to be determined) linear combination of the range of zerothorder 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 + ElamE} )qu — E w‘ + E‘ Zealw;
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 ﬁrstorder 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 hydrogenlike( )+ hydrogen—like( )+ r1'2 So a perturbative treatment might take A A A A 1
0 1
H( ) = thdrogenlike + 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 twoCoulomb 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 551551
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 DOI 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 (SpringerVerlag, 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 singlepoint, 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? Ianm (W) HF/ccpVTZ M HH Distance (angstroms) CHEM 533 Spring '97 the class‘ Cl caiculations on H2 input the data from chem533b.txt (distanceenergy) (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,zi,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 =0752 ang. con = el.l62 Hartree
DIE27.21 =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
DIS27.21 =4.56 EV I
am'“ 12052121450240“ .4 v 3
vib. freq ‘r.:_..  I , __ v=l.47610 Hz .. [B =4_92210 cm“
 ' ll "‘  _
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 HH (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.
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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: 631G (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.486D01
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 = 0.1033869638DOl E2: LO.204O471822DDl
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 Avectors is 0.5266026DOl 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)= 0.31970389D01 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= 5.291772490000D01 E—N=—3.ll7032851892D+00 KE= 9.241504380383D—01 Symmetry AG KE= 9.241504380383001
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
11GINC—UNKISPIRCISD—FCI6—31GIH2PCUSER125—Apr~19970ll# CI 6—316 SPI
lhydrogenlI0,llHIH,l,l.lVersion=4B6Windows—G94RevB.2State=1—SGGIHF:
e1.094808MP2=—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 BornOppenheimer 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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This note was uploaded on 07/17/2008 for the course CHEM 861 taught by Professor Dr.coe during the Fall '04 term at Ohio State.
 Fall '04
 DR.COE
 Atom, Quantum Chemistry

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