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Mixed valence—Rydberg states
Volcme 35, number 4
.CHEhIICAL PHYSICS LETTERS
MfXED VALENCE-RYDBERC Robert J. BUENKER*and Lchrsrrthl fiir Tkoretische
I
December 1975
STATES.
Sigrid D. FEYEXIMHOFF
Cilertzfc der Universitiit Bonn, 53 Bonn, West Germguy
Received 28 August 1975
A survey of recent calculations involving the mixing of Rydbcrg and valence states in the spectra of molecular systems’ is undertaken. It is pointed out that when such states undergo curve crossing with one s-mthcr. minim& energy splittings of 1.O-2.0 CV c;ln occur at the respective SO-SO composition points. Th& significance of such large interactions between valence and low-lying Rydberg states is considered in tegrnj of the properties of the resuking mixed CI states, particularly with reference to the oscillator strengths far the pertinent ground state transitions and also the spatial extension of the corresponding upper orbit&. It is thereupon areed that such mixinys have important consequences in the spectra of a wide variety of systems, including those of 02, ethylene and ethanc.
1. Introduction IMU& of the early theoretical work in molecular spectroscopy has tended to emphasize the role of valence-shell excited states almost to the point of exclusion; this is particularly true of the first PariserParr-Pople calculations on n-electron systems [I]. Yet the existence of another type OF state in molecu1~ systems, quite familiar in atomic spectra, in which the upper orbital exhibits a very expanded charge distribution, has been known for a relatively long time now, dating back to the experimental work of Price [2 J . To a good appro~matio~ the outer clcctron
in such(Rydberg) states can be said to experience the field of a hydrogenic atom since the molecular framework is relatively smali compared to the dimensions of the orbital in which the eIecrron is located, and this characteristic has been shown to lead to the
occurrence of series of such states converging to the limit of ionization in which the electron is completely
removed from the nuclear environment. Despite the obvious distinctions between vdence-shelf. and Rydberg states in molecules it has nevertheiess proven very dificult to c&struct a defmition of either type * Senior U.S. Scientist Awzrdee of the Alexander von Humboldt foundation, on leave from the Dkpz&nent of Chcmiwy, Unip%sity of Nebbrasb, USA.,.
of state which is entirely satisfactory. The basic problem is that ihe supposed dichotomous relationship between valence-shell and Rydberg states is not always valid; there seems to exist. a gray area between the two extremes which has led to no Little controversy in recent years, particularly since the time when ab initio calculations of molecular excited states have been employed
in such investigations
From a theoretical
[3].
point of view the uncertainties
rais%i
by the lack of a univer$ally suitable definitiori of the two general types of states are probably best confronted by asking the question of how strondy pure Rydberg and valence-shell species of the samd symmetry can influence one another when they occur at roughly the same enerB in a given electronic spectrum.
In addition it is useful to consider what the
properties of a nixed state wouid be, that is, one whose wavefunction contains roughly equal contributions from a pure valence state and a pure Rydberg species. In the present paper results,o’ab initio SW. and CI calculations for ths systems O,, ethylene and ethane are considered with an eye toward providing at least partial answers to these two related questions..
2. Mixing of Rydberg a$
Mlence stat& in 02 .-
It is known’-+&a; Ridberg and valence states c&en
415
Volume 36, number 4
CHEMICAL PHYSICS LETTERS
exhibit markedly diffc?ent geometrical characteristics and as a result it is possible that the stability order of such species can EI’~ s@ifrcontly wiih changes in nklear conformation. A good examp!e of such behavior has been found by Mulliken [4] in the spectrum of BH, with a normal valence excited state graduaily being transformed into a pure Rydberg species as the internuclear distance is decreased. Recent calculations on the excited states of 0, provide still another clear instance of such a phenomerion [5,6]. In this case vale’nce states of IT~T: and 430” configuration are found to undergo curve crossings with the n,3p Rydberg species ofthe same symmetry at relatively small bond lengths (1 O-l.2 a). One of the more intcresttig results of this study is the magnitude of the minimal enera splittings calculated for these valence-Rydberg curve crossins at the point in which the adiabatic wavefunction possesses roughly SO-.50 composition of Rydberg and valence-shell character respectively. In the case of the 3A, species this quantity is very sma!l, only a few hundredths of an eV, indicating almost no interaction between the two types of electronic configurations- Yet the 3k; states of the analogous valence (nz$) and Rydberg (np3pn,) configurations do undergo significant interaction, with a minimal splitting between the corresponding adiabatic (CI) curves of 0.98 eV having been calcu!ated [6] (for an R value very close to t_le equilibrium distance of the 3Xf ground state). The mixed valence-Rydbcrg character of the 3AU and 3X; states at this R value is easily seen from the magnitudes of the pertinent CI espansion coefficients. For.the mixing between the 311, states in the same region of the spectrum the situation is somewhat complicated by the fdc:t that both types of configurations have the same fo:m, namely xgno, ; in the valence
state
the outer
ozbital,is
2po*
while
in the
Rydbcrg configuration it is the 3~0, species*. As a result the mixing between Rydberg and valence states can occur at the orbital level and hence it is no longer so clear how to judge Ihe relative contributions of the two type of configurations. Nevertheless the minimal l
Thk distinction is most simply made on the basis of excitation class: in the 3~; Ed 3AU ~~s-es the IWO configurations dtifer by a doubk escitotion, whereas in the 3nu case they differ by only a single excitation.
1 December
i975
splitting between the adiabatic potential curves of the corresponding mixed states still remains a valid rneasure for the degree
of interaction
between
the pure
components of valence and Rydberg type, and the calculations show that this quantity is even larger in the ‘II, c;lse than for the 3Z; species (2.12 eV)*. The conclusion from these 07 calculations is clear. Valence
species
can undergo
a strong
interaction
with
member thereof, with the resulting mixed valence-Rydberg adiabatic states occllrtilg at erzei-gies as nrrdz as 0.5-I. 0 c V (half the minimal splitting alluded to above) below the locctiorz of the comsporrdi~zgpure (unperturbed) stares cotmibuti:~g thereto. In such cases the resulting states are best described as roughly equal mixtures of Rydberg
pure
states,
valence
parricularly
(small
the II = 3
radius)
and pure
Rydberg
(relative-
ly large radius) states. Experimental evidence for such interrctions has been given earlier by Douglas 171. The
remaining
question
is then
how
frequently
strong mixings between valence and Rydberg states occur for other systems. do such
3. Properties of mhed valence-Rydberg
states
such mixed states naturally requires an understanding of how their various properties will be perceived experimentally. Particular attention should be directed to the electronic transition moment (or oscillator strength) and also the degee of spatial Recognition
extension,
since
of
these
quantities
are most
often
used
to attempt to distinguish between Rydberg and valence species in a given electronic spectrum. Quantum mechanically
this objective
can be obtained
ing a simple wavefunction fOi f CR GR, where the subscripts
such states, m, v and R mixed, (pure) valence and (pure) Rydberg respectively. The transition moment from excited state is then given as ($$ed
by zonsider-
$m = C,$, refer to species ground to
$mj =pgm = c”!.$, + =RhR -
(1)
The fvalue for such a transition is thus proportional
t* Use of more diffuse functions in the A0 basis would lower this result somcwha: because of the expected beneficial effect it would hsve on the representation of the upper adiabatic state (but very little on the lower), but the true minimum splitting should still lie in the 1.5-2.0 eV re&ion.
Volume 36, number 4 to the square of this quantity
(assuming
real coeffi-
cien ts):
fp'Xc3L&12+C~lP~RIZ~~C~C,lPpilLL~I- (2) A similar expression can be given for (4) of the upper orbital in such transitions to measure its spatial extension”: (~}m,,,=C,‘(;),iC2R(;)RRC2CvCR(;}FR.
(3)
In the case of the oscillator strength [eq. (2)] one expects the valence term to dominate and hence the fvalue for such a transition is expected to be roughly equal to the fraction cz of the corresponding pure valencef‘value. As usuaI in such situations, however, it is easily possible thar the cross term is eq. (2) will also have an important influence on the final result, either adding to or subtracting from the pure valence contribution. For spatial extension quantities (including the charge density) quite similar conclusions hold except that in this case it is the Rydberg term which dominates; again the main contribution should come from the corresponding diagonal term (with proportionality constant ci) but the cross term can also be a factor which should not be overlooked in this instance. The O2 results provide an instructive example of how these simple ideas work in practice. At R = 2.28 bohr the iowest 3Z; CI state is a roughly 60-40 mixture of valence and Rydbcrg states respectively (based on lc21 results). The transition moment to tfle mixed state has been analyzed in detail in table 3 of ref. [6]. it is found that + (7,~~. element) in this case is 1.13 au compared to the-relatively small value of pfi (xg3px,) of -0.03 au. Despite the large CI coeffic&t for the Rydberg species a secondary valence configuration (30~ + 30, relative to the ground state) is found to fmve a greater effect on the total transition moment pW than the Rydberg contribution. The f’value for the pure valerlce transition (fin: configuration) is calculated to be 0.55 but the co;bined effect of ihe 40% Rydberg character plus the secondary valence species brings this value down to 0.158 for the net oscillator strength of the 3Xi mixed state, which in
l
1 Dzcembcr
CHEhlICAL PHYSICS LETTERS
The coefficients c have the same meaning as before if IJ\, and +R differ by n single excitation. Othenvisc the cxprcssion is somewhat more complicated, but or course similar in structure.
1975
turn agrees quite well with the measured fvaI& for the pertinent experimental (Schumann-Rungej band system [H] of 0.162. Clearly failure to include the significant contribution from the srg3pn, (ae25ruR) Rydberg state in the pertinent wavefunction would lead lo a large overestimation of this quantity. At the same time, however, adding such character inevitably has a strong effect on the charge distribution of this state, causing it to be far more extended in space at this bond distance than would otherwise be expected on the basis of its z:,: leading configuration.
4. ~Wxing between in ethylene
the
‘(~i,$@) and ’ (n,3dn)
states
It was pointed out by Mulliken in 1942 that the *(n,x*) valence configuration in ethylene has the same symmetry as the ‘(n,3dn,Y,) Rydberg state of this system. The fact that this pure 3d Rydberg species should be found at an energy of 8.5-9.0 eV, that is, roughly 1 .O eV above the location of the Franck-Condon maximum in the broad N-V absorption bands associated with the ‘(,,z*) state, however, seemed to rule out the occurrence of any significant
mixing between the Rydberg and valence species in this case. Nevertheless in view of the 0, results discussed in section 2 there is reason IO consider such a possibility more carefully. Indeed when a CI caIcu!ation is carried out in which both the ‘(z,x*) vaIence and ‘(~,3dx) Rydberg species are present, the results show two adiabatic states !ocated at 8.1 and 9.6 eV respectively [9], that is, with an energy splitting of 1.5 eV quite similar to what is calculated for the 3 Cu (0.98 eV) and ‘II” (2.12 eV) mixed valenceRydberg states of 0, at their respective 50-50 COWposition points. The two ethylene configurations in question differ by a single excitation and hence, as before with the O2 ‘IIu states, it is somewhat difficult to distinguish between them since mixing can already- occur at the orbital (MO or NO) level. This identification problem can be virtually eliminated, however, by employing an (orthogonal) basis set for the CI calculations in which the Rydberg 3dT A0 is prevented in so far as possible from mixing with the valence-type T* MO’s_ This condition
is achieved
with the ground
to a quite
good
approximation
state NO basis for ethylene,
as can 417
Vohmc
36, number
Table 1 CaIcuIeted transition
4
moment
CHEMICAL PHYSICS LETTERS
I December
1975
data for the ‘(n;n*) state in ethylenea) --___
Characteristics of lower of and upper (r) orbital _--
Jir of upper orbital
Dq
Zij
Dpfj
f value for pure state
ilJlpl+ z;/tp*n* ni3dn*z x/n* polarization
0.506 0.267 0.105 0.561
0.6920 -0.5630 0.8942 0.1399
I .2878 -0.5831 0.1795 -0.0583
0.8911 0.3283 0.1605 -0.0082
0.661 0.135 0.013 0.001
species
partial sum of Df (valence) i (Rydberg) partial sum of Dri ____-
____---
0.82 0.80
_.__ _.__._-__I_
a) Listed arc theCoulomb intepals Jir as a measure for the compactness of the upper orbital; in addition matrix elements ZQ = J$+rbidr between upper and lower orbital, the. corresponding fist order density f v&e for the pure S+Qtes, where := -J [:~D$cF]2 rlE (’m each of the four cases cited D$cF = 2”?).
be judged from the s&-energies (Coulomb integrals J;r) of the VM&RIS E* species; a pure 3drr A0 has aJii value of 0.091 and this compares quite favorably with that of the 3b2a NO (0.105) but with none of the other three orbitals or‘ this symmetry (table I)_ A CI for the lowest I(rr,rr*) state using these NO’s leads to rather strcng occupation of all four of the b,, species, thereby indicating a high decree of mixing between the component Rydbars and valence states; on the basis nf the squares 0:‘ the corresponding CI expansion coi%cients one calcuiates that the Rydberg (x,3&r) configuration thereof is present to the extent of 4375, with the remaining contributions falling to the other three ‘(n,n+)-type valence-shell species as indicated in table l*_ Despite the relatively large contribution from the (x,3dir) Rydberg configuration, howe%, it is still clear from table 1 that the resultant mixed state retains si@ficant valence-shell characteristics. -4 breakdown of the calculated transition moment data to the state at 8.1 eV shows (table I) that the two ‘(lr,l;*) ‘valence configurations contribute I.22 au while their Rydberg counterpart :tdds only 0.16 au thereto. -4s usual the presence of the. Rydberg configuration in the mhed state has a damping effect on the totalf value for the transition, but the net resuit is still relatively large (0.291, and most importantly, is found to be in good agreement with the measured value for this quantity of 0.3’4 [lo]. Failure to include the Rydberg 3dn Functions in the basis not orly ieads to
the transition moment matrix value Dq and the
a marked increase in the enerw of the lowest calculated ‘(gi,zr*) state 00 8.9 eV [3]).but also to a sign.ifIcant overestimation of the associated f value (0.485, table 2) compared to experiment. This effect is easily understood from the fact that while the most valence-like l(l;,n*) configuration in this calculation corresponds to a purefvaiue of 0.661 (table I), the analogous result for the Rydberg (i;,3dri) transition is only 0.013. The effect of the strong mixing between valence and Rydberg configurations is also apparent from consideration of other properties, particularly the spatial expansion of the upper orbital in the assocl.?.:ed transition. Thus while a pure valence rr* MO shows a value for (~2~)of only 3.88 au (from a calculation for the 3(n,n*) state of this system), the analogous result for the Rydberg 3dn A0 is 53.6 au (table 2); the COP responding NO in the mixed state falls roughly midway between these values at 24.3 au, The interesting point, however, is that although both the (z”> andf Table 2 Comparison of characteristic datr? for pure valence and pure Rydberg states with those of the cakulated t(r,x*) state in ethylene a) -State
Jj#KI)
I
(z2>
:~~,~~~Ydber~ *
0.09 1 0.148 0.403 0.506
0.015 0.293 0.482
53.6 24.3 b) 3.88 -
3(3:~*f valence ‘(n,x*) without 3 drr conttibut;on1
* More extensive calculations
have shown that the lowest energy for states of this symmetry is obtained when the amount .. of Rydbtrg fn,3dn) character is in~wscd to rou@y 48% PI.
‘418
a) For notation
see table 1; (z*) = (oilz*11$$ in au. bI Vaiues from NO basis,
Volume 36, number 4
CHEMICAL THYSICS LETTERS
value results for the mixed state are seen to be quite consistent with a nearly SO-SO composition, the very magnitudes of these quantities tend IO obscure this simple fact. For exampie, a (z")value of 24.3 au is not obviously consistent with anything but a Rydberg excited state (see also the corresponding charge density contours in fig. 4 of ref. [ 1 I]); nevertheless the determination of anfvalue fcr the associated transition of 0.29 certainly makes it difficult to identify the upper state as being anything other than a valence species. In view of these results it does not really seem so surprising that the appearance of the ethylene spectrum in the N-V region is not greatly altered in condensed phase experiments 1121; such behavior is bown to be uncharacteristic of pure Rydberg transitions, but when the upper state consists of 50-60% valence character the situation is far less clear. On the other hand the assignment of the upper state in question as a strongly mixed species is quite consistent with the findings of Krauss and Mielczarek 1131 who do detect a significant amount of Rydberg character for the upper state on the basis of their measurements of the angular dependence of generalized oscillator strengths in inelastic electron scattering.
5. Mixing with higher Rydberg
states
In the previous sections emphasis has been placed on the properties of the lowest-lying CI state resulting from a nearly equal mixing of pure valence and Rydberg species. Clearly the occurrence of such a mixed state implies that the pure unperturbed components involved must possess roughly equal enera, and this observation raises questions about the composition of the cafijugale CI state. Does the existence of a SO-50 mixed valence-Rydberg state at low energy, for example, guarantee that a second state of very similar (i.e., also about 50-50) composition be found above the location of the unperturbed states inter-
acting in this process? The answer to this question appears to be an unqualified no, and the reason for this somewhat surprising situation is clearly seen to lie in the fact that the conjugate state in such an interaction inevitably can be further mixed with still other higher-order Rydberg species; in particuiar in the ethylene ‘(z,z”) case the ‘(z,4dn) is capable of making a strong inter-
1 December 19’55
action with the corljlcgaie (higher-lying) mixed state resulting from the (iT,Ti*)-(Ti,3dz) mixing. To demonstrate this effect in calculations it is obviously necessary to include functions in the A0 basis which are adequate for the representation of both the II = 3 and II = 4 Rydberg states. If only one diffuse orbital is available (with optimizedwponent 01= 0.02 [l I]) only the ‘(z,3dz) pure species can be represented and hence Ci calculations at this level do produce a second mixed state with roughly SO-50 valence-(r,3dr) composition; but the energy associated with this artificial state (9.6 eV) is well above the expected location of the higher-lying Rydberg species and the fvalue associated with the corresponding transition from the ground state (0.23 [9]) is much too large to be identified with any observed absorption system in the C,H, spectrum. Adding a second diffuse species to the A0 basis with still smaller exponent (0 = 0.014) to represent the II = 4 member leads to a substantial lowering in the energy of this (second) mixed ‘(n,n*)-Rydberg state to 8.97 e\l, for ~!/practicalpurposes without altekg the stability of the lower-energy species discussed in section 5. Furthermore, the oscillator strength calculated for the corresponding transition shows nearly a fourfold decrease (to 0.07) compared to its artificial value in the more restrictive calculation with only one such diffuse Rydberg-type basis function*. Both the energy location and thefvalue associated with this second state in its proper representatiorz are in good agreement with Wilkinson’s experimental results for the R(Iv) bands in ethylene [14], w~hich are assigned in the original reference to a (ir,ndn) transition with possible pertrubatims from the lower-lying ‘(n,n*) r!aletzce state. Realization that a pure (?r,4dn) Rydberg transition should show anfvalue somewhat m the 0.005-0.01 range emphasizes that the state in question, despite its heavily_ diffkse conzpositiorz, still contains a significant amount of of valence (zJ*) c h aracter; the calculatedfvalue 0.07 would correspond to IO-15% valence state mhing in this case. Clearly this process of mixing the 1(.rr,.ir*> valence state with ‘(rr,ndx) Rydberg species will continue up
l
The prcscnce of the extra diffuse function does not change the transition energy to the hoest l(~,z*) state (8.1 eV), and it causa
the correspondingfvzlue
to decrease by only
0.006.
419
V&time 36, number
4
CHEMICAL PHYSICS LETFERS
1
Deceinber
1975
through higher quar: turn numbers. The j’value for the rid-d l(~,niL)--l (n,Jldn) mixed state in the calculation ,witt-two diffuse AO’s is s!iU too large (0.10-o. I5 [9j) tc be credibly identified with a [rue experimen-
in which it is not. three distinct cases can be distinguished according to the second key factor, namely the enera separation between the unperturbed (pure)
tal transition. Addition of more diffuse AO’s with stiU 2mzLler exponents to the basis set will clearly
In case I, in which the valence state lies well below all members of the various Rydberg series, one expects essentially no mixing effect (fig. I). Experimentally it is known [ 1 S] that Rydberg states possess term
Rydberg and valence levels.
result in a litrC/ler pmtitioni~tlg of‘rllis residrrai (n,n*) valehce4rell charac:er anmlg the higher-order mem hers of the ‘( z,ndz) Ryclbbcr,- mmuf~~d until cventually the iirnit of ionization is reached. Since the capaciti of a pure Rydberg state to mix in valence character should decrease I‘airly quickly as the size of the upper orbital increases (thereby causing the hamiltonian interactix mztri:; element with the pertinent valence state to become even smaller) it seems clear that after II Increases to fairly large values the deviations of the various Rydberg lines from their normal (unperturbed) pattern will all but disappear.
6. Mode! for mixing
of valence
The probability
of signilicant
valence
separate
and Rydberg
states
and Rydberg mixing
clcnrly
values
states
Ion
--
W, I n-_5 MCI
I
.’
426.,-
Case
* That is, the minimal energy splitting bctwcen upper end lower ,nixcd states xe found to vary from 0.0 to 1.0 eV. As discussed in section 5, vnlucs of MvRshould dccrcusc krly quicklv with the Rydberg quantum number II. ** There is some theorcticd support for the idea that H,R tvill bc largest
\vhen
H
ton
I’ i
Case Iii
,
m
I I
_----_-
PI<
differ
by only
ValFAlCe Ion, n-5
etc i
n=L m,red
i
I
Ryd-
krg
n=3 1
valencel
I
I I
t t
diagram showing the interaction
states
but in view of the results for the 3E; smtcs in O2 (section 2) it is clear that signiiicant cffccts can also occur when a double excitation separates the unperturbed slates.
---
the unperturbed
a single cscitation
I ! I I I ; , v z!cz!z~~{xK-=
n=3
IP) in the
I
I
n=L
-Valema
Fip. i. Schematic
I
rhe corresponding
on two
hamiltonian matrix element H,R connecting the VHlence species with a given Rydberg state. The calculitions for 0, and ethylene discussed above indicate that this quantity can lie anywhere in the range from 0.0 to 1.0 eV* for the lowest member of a Rydberg series. if HVR is quite small ‘very little mixing will occur, but for the (apparently numerous) instances
Case
below
between
depends
‘The first is the magnirudc of the
factors.
(energies
3.0-3.5 eV range for 3s, 2.2 to 2.8 eV for 3p and 1 .O to 1.8 eV for 3d members respectively. Thus if the maximum absorpticn in a given band system lies in the order of 7.0 eV below the expected location of the lowest-lyingpltrc Rydberg species of the same mixing is very unlikely. Good examples symmetry*, for this situation are supplied by the 3(n,r*) and the 3T1(n,+) valence siates in ethylene [l I] and formaldehyde [ 16,171 respectively. The minimum IP in ethylene is 10.51 eV and hence the lowest-lying Rydberg stare of the required symmetry in this instance, namely 3(7r,3d=), should be found no lower than roughly 8.5 eV. At an energ’ of 4.32 eV [I 81
between
valence and Rydbeg
states in
the
three cases discussed
in the text.
Volume 36, number 4
the 3 (z,fl*) valence trum
CHEMICAL
state is thus too low in the spec-
to be significantly
influenced
by the Rydberg
states of ethyiene. Similarly the 3~1(n,~*) states of HzCO found at 3.4 and 3.8 eV respectively are much too stable to interact noticeably with the pertinent 311(n,3dn) species (the IP for the n orbital is 10.86eV). Jn all these examples calculations including diffuse (Rydberg-type)
basis functions
very small tendency any degree
have shown
for these valence
of Rydberg
only
a
states to assume
character_
the situation in which the pertinent valence state roughly coincides in energy with the low-lying members of a Rydberg series of like symmetry (fig 1). Under these conditions the lowest adiabatic (CI) state can be fotind considerably MOW (by HVR) the expected location of the (pure) lowest-lying Rydberg state of this symmetry. In addition it is very likely that a good portion of the (unperturbed) valence character will be partitioned among the various higfler-order members of the same Rydberg series, as discussed in section 5. The often treated ‘(7,7-r*) absorption in C,H, and the 3Z, and 311,, - X transitions in O7 near 1.2 A appear to fall into this category* and numerous other examples which aiso fit this description are known, such as the ‘(FJ*) transitions in diimide and acetylene, for example [20,2 1 ] . The third and final case then involves the occurrence of a valence state which lies beyond tfle location of the minimum IP of the system (fig. 1). In this situation it is possible to have a large (usually infinite) number of Rydberg Sates lying below the valence species. The similarity with case I is obvious; if the value of HVR is quite large in a given instance some mixing between the Rydberg (especially the lower members) and the valence states is nevertheless possible. A good example for this behavior has been found in the study of the ethane spectrum. The location of the ‘(a,~‘) state in this system is found to vary depending on the number of diffuse (3pa) AO’s included in the A0 basis, thereby indicating that some interaction with the Rydberg states of like symCase 11 then
* The pertinent
involves
(unperturbed) 3(ng,3p) Rydberg species in O2 should have vertical excitation energies of 9.2-9.6 eV [5,6] while the pure ‘(;;,3dn) state in ethylene should tie in the 8.7 eV region [191,i.e. only1.OCV above the Frznck-Condon maximum in the V-N absorption system.
PHYSICS LIXTERS
1
December
1973
metry lying below the valence state is in fact occurring. This point is emphasized by the unusually largefvalue (0.15) calculated for the ‘(cr,3po) transition in ethane at 9.5 eV [23]. Because of the very high oscillator strength for the pure ‘(~,a”) valence-shell transition (probably 0.8 to 1 .O) even a small admixture of such valence character into a normal 3~0 Rydberg wavefunction can have a noticeabk effect on thef‘value of the corresponding transition to the resultant (weakly) mixed state. The same effect appears to be present to a somewhat lesser degree in the ‘(u,4po) transition, for which an fva!ue of 0.07 has been calculated (using gaussian exponents of 0.017 and 0.0!3 to represent the first two Rydberg orbitals). At the same time the calcs!ations indicate that the valence ’ (a,~*) state should be found at ro,ughly 16 eV with an /‘value of from 0.65 to 0.80, all of which results in turn agree reasonably well with the measured absorption and electron impact spectra of ethane [23,24], including the existence of a broad maximum in the continuum The indication is thus that even at 125000 cm-l. when a valence state is well separated from all Rydberg levels, a mixing effect can still be observed if bothH,, as well as the fvalue for the pertinent valence rransition are quite large (and the same would seem to hold in an analogous manner for case 1); on rhe other hand, from the standpoint of the spatial exterzsiorz of the corresponding upper orbital little or no indication of such a weak m&.ing is noticeable [22].
Acknowledgement
The authors wish to thank Professor C. Sandorfy and Dr. H. Lefebvre for various: stimulating discussions on the subject of Rydberg versus valence states and the interaction of corresponding potential surfaces. The services and computer time made available by the University of Bonn Computational Center are gatzklly acknowledged.
References [ 11 R. Pariser 2nd R.G. Parr, J. Chem. Phys. 21 (1953) 466, 767; J.A. Pople, Trans. Faraday Sot. 49 (1953) 137s. [2] W.C. Price, Phys. Rev. 47 (1935) 444; 5. Chem. Phys. 3 (1935) 256.
421
Volume 36, numbqr 4
CHEMICAL
PHYSICS LETTERS
[ 3 ] P.E. Phihipson and R.S. lviullikea, J. Chem. Phys. 28 (1958) 1248. (41 R.S. XIuU;:kcn, Inrern. I. Quantum Chcm. 5s (1971) 83. (51 R.J. Buenker and S.D. Peyerimhof’f, Chem. Phys. 8 (I 975) 324. (61 R.J. Buenker and S.D. Pcyerimhoff, Chem. Phys. Lctters 34 (1975) 225. (7 1 A.E. Douglas, in: Chemical spectroscopy and photochemistry in the vacmlm ultraviolet, eds. C. Sandorfv, PJ. Ausloos and X1-B. Robin (Reidel, Dordrecht. 1974). (B] RN. Huebner, R,J. Celottc, S.R. hficlczarek ;Ind C.E. Kuystt, Bull. Am. Phys. Sot. Nov. 1974, DCS, p. 1194. [9] ,R.J. Buenker and SD. Pcyerimhoff, Chem. Phys. 9 (1975) 75. [lo] A.J. Merer and R.S. hlulliken, Chem. Rev. 69 (1969) 639. [Ill R.J. Bucnkcr, S.D. Peyerimhoff and W.E. Kammcr, J. Chem. Phys. 55 (1971) 814. !12] E. hiiron, 9. Rx and _I. Jortner, J. Chem. Phys. 56 (1972) 5265. !131 M. Krauss and S.R. Mielczuek, J. Chem, Phys. 51
(1969) 5241; ILLRobin, Higher excited states of polyatomic cults, Vols.
1975).
.422
:_
l/2 (Ac;ldemic
molePress, New York, 1974/
1 December 1975
1141 P.G. Wilkinson, Can. J. Phys. 34 (1956) 643. [IS] C. Sandorfy. Lecture given at the VII International Conference of Photochemistry, Jerusalem, Israel, August 1973. [ 161 R.J. Buenker and SD. Peyerimhoff, J. Chem. Phys. 53 (1970) 1368. [ 171 J.L. Written and Xl. Hackmeyer, J. Chem. Phys. 51 (1969) 5584. [ 181 J.H. hloore Jr., J. Phys. Chem. 76 (1972) 1130. [19] U. Fischbach, R.J. Buenker and S.D. Peyerimhoff, Chem. Phys. 5 (1974) 265. (201 R. Vnsudevan, S.D. Peyerimhoff, R.J. Buenker and W.E. Kammer, Chem. Phys. 7 (1975) 187. 1211 W.E. Knmmcr, Chem. Phys. 5 (1974) 408. [22] R.J. Buenker and S.D. Peycrimhoff, Chem. Phys. 8 (1975) 56. [23] C. Sandorfy, in: Chemical spectroscopy and photochemistry in the vacuum ultraviolet, eds. C. Sandorfy, P-r_ Ausloos and M.B. Robin (Reidel, Dordrecht, 1974). [24] E.N. Ussettrc, A. Skerbelc and M-A. Dillon, J. Chem. Phys. 46 (1967) 4536;48 (1968) 539; 49 (1968) 2382.
.CHEhIICAL PHYSICS LETTERS
MfXED VALENCE-RYDBERC Robert J. BUENKER*and Lchrsrrthl fiir Tkoretische
I
December 1975
STATES.
Sigrid D. FEYEXIMHOFF
Cilertzfc der Universitiit Bonn, 53 Bonn, West Germguy
Received 28 August 1975
A survey of recent calculations involving the mixing of Rydbcrg and valence states in the spectra of molecular systems’ is undertaken. It is pointed out that when such states undergo curve crossing with one s-mthcr. minim& energy splittings of 1.O-2.0 CV c;ln occur at the respective SO-SO composition points. Th& significance of such large interactions between valence and low-lying Rydberg states is considered in tegrnj of the properties of the resuking mixed CI states, particularly with reference to the oscillator strengths far the pertinent ground state transitions and also the spatial extension of the corresponding upper orbit&. It is thereupon areed that such mixinys have important consequences in the spectra of a wide variety of systems, including those of 02, ethylene and ethanc.
1. Introduction IMU& of the early theoretical work in molecular spectroscopy has tended to emphasize the role of valence-shell excited states almost to the point of exclusion; this is particularly true of the first PariserParr-Pople calculations on n-electron systems [I]. Yet the existence of another type OF state in molecu1~ systems, quite familiar in atomic spectra, in which the upper orbital exhibits a very expanded charge distribution, has been known for a relatively long time now, dating back to the experimental work of Price [2 J . To a good appro~matio~ the outer clcctron
in such(Rydberg) states can be said to experience the field of a hydrogenic atom since the molecular framework is relatively smali compared to the dimensions of the orbital in which the eIecrron is located, and this characteristic has been shown to lead to the
occurrence of series of such states converging to the limit of ionization in which the electron is completely
removed from the nuclear environment. Despite the obvious distinctions between vdence-shelf. and Rydberg states in molecules it has nevertheiess proven very dificult to c&struct a defmition of either type * Senior U.S. Scientist Awzrdee of the Alexander von Humboldt foundation, on leave from the Dkpz&nent of Chcmiwy, Unip%sity of Nebbrasb, USA.,.
of state which is entirely satisfactory. The basic problem is that ihe supposed dichotomous relationship between valence-shell and Rydberg states is not always valid; there seems to exist. a gray area between the two extremes which has led to no Little controversy in recent years, particularly since the time when ab initio calculations of molecular excited states have been employed
in such investigations
From a theoretical
[3].
point of view the uncertainties
rais%i
by the lack of a univer$ally suitable definitiori of the two general types of states are probably best confronted by asking the question of how strondy pure Rydberg and valence-shell species of the samd symmetry can influence one another when they occur at roughly the same enerB in a given electronic spectrum.
In addition it is useful to consider what the
properties of a nixed state wouid be, that is, one whose wavefunction contains roughly equal contributions from a pure valence state and a pure Rydberg species. In the present paper results,o’ab initio SW. and CI calculations for ths systems O,, ethylene and ethane are considered with an eye toward providing at least partial answers to these two related questions..
2. Mixing of Rydberg a$
Mlence stat& in 02 .-
It is known’-+&a; Ridberg and valence states c&en
415
Volume 36, number 4
CHEMICAL PHYSICS LETTERS
exhibit markedly diffc?ent geometrical characteristics and as a result it is possible that the stability order of such species can EI’~ s@ifrcontly wiih changes in nklear conformation. A good examp!e of such behavior has been found by Mulliken [4] in the spectrum of BH, with a normal valence excited state graduaily being transformed into a pure Rydberg species as the internuclear distance is decreased. Recent calculations on the excited states of 0, provide still another clear instance of such a phenomerion [5,6]. In this case vale’nce states of IT~T: and 430” configuration are found to undergo curve crossings with the n,3p Rydberg species ofthe same symmetry at relatively small bond lengths (1 O-l.2 a). One of the more intcresttig results of this study is the magnitude of the minimal enera splittings calculated for these valence-Rydberg curve crossins at the point in which the adiabatic wavefunction possesses roughly SO-.50 composition of Rydberg and valence-shell character respectively. In the case of the 3A, species this quantity is very sma!l, only a few hundredths of an eV, indicating almost no interaction between the two types of electronic configurations- Yet the 3k; states of the analogous valence (nz$) and Rydberg (np3pn,) configurations do undergo significant interaction, with a minimal splitting between the corresponding adiabatic (CI) curves of 0.98 eV having been calcu!ated [6] (for an R value very close to t_le equilibrium distance of the 3Xf ground state). The mixed valence-Rydbcrg character of the 3AU and 3X; states at this R value is easily seen from the magnitudes of the pertinent CI espansion coefficients. For.the mixing between the 311, states in the same region of the spectrum the situation is somewhat complicated by the fdc:t that both types of configurations have the same fo:m, namely xgno, ; in the valence
state
the outer
ozbital,is
2po*
while
in the
Rydbcrg configuration it is the 3~0, species*. As a result the mixing between Rydberg and valence states can occur at the orbital level and hence it is no longer so clear how to judge Ihe relative contributions of the two type of configurations. Nevertheless the minimal l
Thk distinction is most simply made on the basis of excitation class: in the 3~; Ed 3AU ~~s-es the IWO configurations dtifer by a doubk escitotion, whereas in the 3nu case they differ by only a single excitation.
1 December
i975
splitting between the adiabatic potential curves of the corresponding mixed states still remains a valid rneasure for the degree
of interaction
between
the pure
components of valence and Rydberg type, and the calculations show that this quantity is even larger in the ‘II, c;lse than for the 3Z; species (2.12 eV)*. The conclusion from these 07 calculations is clear. Valence
species
can undergo
a strong
interaction
with
member thereof, with the resulting mixed valence-Rydberg adiabatic states occllrtilg at erzei-gies as nrrdz as 0.5-I. 0 c V (half the minimal splitting alluded to above) below the locctiorz of the comsporrdi~zgpure (unperturbed) stares cotmibuti:~g thereto. In such cases the resulting states are best described as roughly equal mixtures of Rydberg
pure
states,
valence
parricularly
(small
the II = 3
radius)
and pure
Rydberg
(relative-
ly large radius) states. Experimental evidence for such interrctions has been given earlier by Douglas 171. The
remaining
question
is then
how
frequently
strong mixings between valence and Rydberg states occur for other systems. do such
3. Properties of mhed valence-Rydberg
states
such mixed states naturally requires an understanding of how their various properties will be perceived experimentally. Particular attention should be directed to the electronic transition moment (or oscillator strength) and also the degee of spatial Recognition
extension,
since
of
these
quantities
are most
often
used
to attempt to distinguish between Rydberg and valence species in a given electronic spectrum. Quantum mechanically
this objective
can be obtained
ing a simple wavefunction fOi f CR GR, where the subscripts
such states, m, v and R mixed, (pure) valence and (pure) Rydberg respectively. The transition moment from excited state is then given as ($$ed
by zonsider-
$m = C,$, refer to species ground to
$mj =pgm = c”!.$, + =RhR -
(1)
The fvalue for such a transition is thus proportional
t* Use of more diffuse functions in the A0 basis would lower this result somcwha: because of the expected beneficial effect it would hsve on the representation of the upper adiabatic state (but very little on the lower), but the true minimum splitting should still lie in the 1.5-2.0 eV re&ion.
Volume 36, number 4 to the square of this quantity
(assuming
real coeffi-
cien ts):
fp'Xc3L&12+C~lP~RIZ~~C~C,lPpilLL~I- (2) A similar expression can be given for (4) of the upper orbital in such transitions to measure its spatial extension”: (~}m,,,=C,‘(;),iC2R(;)RRC2CvCR(;}FR.
(3)
In the case of the oscillator strength [eq. (2)] one expects the valence term to dominate and hence the fvalue for such a transition is expected to be roughly equal to the fraction cz of the corresponding pure valencef‘value. As usuaI in such situations, however, it is easily possible thar the cross term is eq. (2) will also have an important influence on the final result, either adding to or subtracting from the pure valence contribution. For spatial extension quantities (including the charge density) quite similar conclusions hold except that in this case it is the Rydberg term which dominates; again the main contribution should come from the corresponding diagonal term (with proportionality constant ci) but the cross term can also be a factor which should not be overlooked in this instance. The O2 results provide an instructive example of how these simple ideas work in practice. At R = 2.28 bohr the iowest 3Z; CI state is a roughly 60-40 mixture of valence and Rydbcrg states respectively (based on lc21 results). The transition moment to tfle mixed state has been analyzed in detail in table 3 of ref. [6]. it is found that + (7,~~. element) in this case is 1.13 au compared to the-relatively small value of pfi (xg3px,) of -0.03 au. Despite the large CI coeffic&t for the Rydberg species a secondary valence configuration (30~ + 30, relative to the ground state) is found to fmve a greater effect on the total transition moment pW than the Rydberg contribution. The f’value for the pure valerlce transition (fin: configuration) is calculated to be 0.55 but the co;bined effect of ihe 40% Rydberg character plus the secondary valence species brings this value down to 0.158 for the net oscillator strength of the 3Xi mixed state, which in
l
1 Dzcembcr
CHEhlICAL PHYSICS LETTERS
The coefficients c have the same meaning as before if IJ\, and +R differ by n single excitation. Othenvisc the cxprcssion is somewhat more complicated, but or course similar in structure.
1975
turn agrees quite well with the measured fvaI& for the pertinent experimental (Schumann-Rungej band system [H] of 0.162. Clearly failure to include the significant contribution from the srg3pn, (ae25ruR) Rydberg state in the pertinent wavefunction would lead lo a large overestimation of this quantity. At the same time, however, adding such character inevitably has a strong effect on the charge distribution of this state, causing it to be far more extended in space at this bond distance than would otherwise be expected on the basis of its z:,: leading configuration.
4. ~Wxing between in ethylene
the
‘(~i,$@) and ’ (n,3dn)
states
It was pointed out by Mulliken in 1942 that the *(n,x*) valence configuration in ethylene has the same symmetry as the ‘(n,3dn,Y,) Rydberg state of this system. The fact that this pure 3d Rydberg species should be found at an energy of 8.5-9.0 eV, that is, roughly 1 .O eV above the location of the Franck-Condon maximum in the broad N-V absorption bands associated with the ‘(,,z*) state, however, seemed to rule out the occurrence of any significant
mixing between the Rydberg and valence species in this case. Nevertheless in view of the 0, results discussed in section 2 there is reason IO consider such a possibility more carefully. Indeed when a CI caIcu!ation is carried out in which both the ‘(z,x*) vaIence and ‘(~,3dx) Rydberg species are present, the results show two adiabatic states !ocated at 8.1 and 9.6 eV respectively [9], that is, with an energy splitting of 1.5 eV quite similar to what is calculated for the 3 Cu (0.98 eV) and ‘II” (2.12 eV) mixed valenceRydberg states of 0, at their respective 50-50 COWposition points. The two ethylene configurations in question differ by a single excitation and hence, as before with the O2 ‘IIu states, it is somewhat difficult to distinguish between them since mixing can already- occur at the orbital (MO or NO) level. This identification problem can be virtually eliminated, however, by employing an (orthogonal) basis set for the CI calculations in which the Rydberg 3dT A0 is prevented in so far as possible from mixing with the valence-type T* MO’s_ This condition
is achieved
with the ground
to a quite
good
approximation
state NO basis for ethylene,
as can 417
Vohmc
36, number
Table 1 CaIcuIeted transition
4
moment
CHEMICAL PHYSICS LETTERS
I December
1975
data for the ‘(n;n*) state in ethylenea) --___
Characteristics of lower of and upper (r) orbital _--
Jir of upper orbital
Dq
Zij
Dpfj
f value for pure state
ilJlpl+ z;/tp*n* ni3dn*z x/n* polarization
0.506 0.267 0.105 0.561
0.6920 -0.5630 0.8942 0.1399
I .2878 -0.5831 0.1795 -0.0583
0.8911 0.3283 0.1605 -0.0082
0.661 0.135 0.013 0.001
species
partial sum of Df (valence) i (Rydberg) partial sum of Dri ____-
____---
0.82 0.80
_.__ _.__._-__I_
a) Listed arc theCoulomb intepals Jir as a measure for the compactness of the upper orbital; in addition matrix elements ZQ = J$+rbidr between upper and lower orbital, the. corresponding fist order density f v&e for the pure S+Qtes, where := -J [:~D$cF]2 rlE (’m each of the four cases cited D$cF = 2”?).
be judged from the s&-energies (Coulomb integrals J;r) of the VM&RIS E* species; a pure 3drr A0 has aJii value of 0.091 and this compares quite favorably with that of the 3b2a NO (0.105) but with none of the other three orbitals or‘ this symmetry (table I)_ A CI for the lowest I(rr,rr*) state using these NO’s leads to rather strcng occupation of all four of the b,, species, thereby indicating a high decree of mixing between the component Rydbars and valence states; on the basis nf the squares 0:‘ the corresponding CI expansion coi%cients one calcuiates that the Rydberg (x,3&r) configuration thereof is present to the extent of 4375, with the remaining contributions falling to the other three ‘(n,n+)-type valence-shell species as indicated in table l*_ Despite the relatively large contribution from the (x,3dir) Rydberg configuration, howe%, it is still clear from table 1 that the resultant mixed state retains si@ficant valence-shell characteristics. -4 breakdown of the calculated transition moment data to the state at 8.1 eV shows (table I) that the two ‘(lr,l;*) ‘valence configurations contribute I.22 au while their Rydberg counterpart :tdds only 0.16 au thereto. -4s usual the presence of the. Rydberg configuration in the mhed state has a damping effect on the totalf value for the transition, but the net resuit is still relatively large (0.291, and most importantly, is found to be in good agreement with the measured value for this quantity of 0.3’4 [lo]. Failure to include the Rydberg 3dn Functions in the basis not orly ieads to
the transition moment matrix value Dq and the
a marked increase in the enerw of the lowest calculated ‘(gi,zr*) state 00 8.9 eV [3]).but also to a sign.ifIcant overestimation of the associated f value (0.485, table 2) compared to experiment. This effect is easily understood from the fact that while the most valence-like l(l;,n*) configuration in this calculation corresponds to a purefvaiue of 0.661 (table I), the analogous result for the Rydberg (i;,3dri) transition is only 0.013. The effect of the strong mixing between valence and Rydberg configurations is also apparent from consideration of other properties, particularly the spatial expansion of the upper orbital in the assocl.?.:ed transition. Thus while a pure valence rr* MO shows a value for (~2~)of only 3.88 au (from a calculation for the 3(n,n*) state of this system), the analogous result for the Rydberg 3dn A0 is 53.6 au (table 2); the COP responding NO in the mixed state falls roughly midway between these values at 24.3 au, The interesting point, however, is that although both the (z”> andf Table 2 Comparison of characteristic datr? for pure valence and pure Rydberg states with those of the cakulated t(r,x*) state in ethylene a) -State
Jj#KI)
I
(z2>
:~~,~~~Ydber~ *
0.09 1 0.148 0.403 0.506
0.015 0.293 0.482
53.6 24.3 b) 3.88 -
3(3:~*f valence ‘(n,x*) without 3 drr conttibut;on1
* More extensive calculations
have shown that the lowest energy for states of this symmetry is obtained when the amount .. of Rydbtrg fn,3dn) character is in~wscd to rou@y 48% PI.
‘418
a) For notation
see table 1; (z*) = (oilz*11$$ in au. bI Vaiues from NO basis,
Volume 36, number 4
CHEMICAL THYSICS LETTERS
value results for the mixed state are seen to be quite consistent with a nearly SO-SO composition, the very magnitudes of these quantities tend IO obscure this simple fact. For exampie, a (z")value of 24.3 au is not obviously consistent with anything but a Rydberg excited state (see also the corresponding charge density contours in fig. 4 of ref. [ 1 I]); nevertheless the determination of anfvalue fcr the associated transition of 0.29 certainly makes it difficult to identify the upper state as being anything other than a valence species. In view of these results it does not really seem so surprising that the appearance of the ethylene spectrum in the N-V region is not greatly altered in condensed phase experiments 1121; such behavior is bown to be uncharacteristic of pure Rydberg transitions, but when the upper state consists of 50-60% valence character the situation is far less clear. On the other hand the assignment of the upper state in question as a strongly mixed species is quite consistent with the findings of Krauss and Mielczarek 1131 who do detect a significant amount of Rydberg character for the upper state on the basis of their measurements of the angular dependence of generalized oscillator strengths in inelastic electron scattering.
5. Mixing with higher Rydberg
states
In the previous sections emphasis has been placed on the properties of the lowest-lying CI state resulting from a nearly equal mixing of pure valence and Rydberg species. Clearly the occurrence of such a mixed state implies that the pure unperturbed components involved must possess roughly equal enera, and this observation raises questions about the composition of the cafijugale CI state. Does the existence of a SO-50 mixed valence-Rydberg state at low energy, for example, guarantee that a second state of very similar (i.e., also about 50-50) composition be found above the location of the unperturbed states inter-
acting in this process? The answer to this question appears to be an unqualified no, and the reason for this somewhat surprising situation is clearly seen to lie in the fact that the conjugate state in such an interaction inevitably can be further mixed with still other higher-order Rydberg species; in particuiar in the ethylene ‘(z,z”) case the ‘(z,4dn) is capable of making a strong inter-
1 December 19’55
action with the corljlcgaie (higher-lying) mixed state resulting from the (iT,Ti*)-(Ti,3dz) mixing. To demonstrate this effect in calculations it is obviously necessary to include functions in the A0 basis which are adequate for the representation of both the II = 3 and II = 4 Rydberg states. If only one diffuse orbital is available (with optimizedwponent 01= 0.02 [l I]) only the ‘(z,3dz) pure species can be represented and hence Ci calculations at this level do produce a second mixed state with roughly SO-50 valence-(r,3dr) composition; but the energy associated with this artificial state (9.6 eV) is well above the expected location of the higher-lying Rydberg species and the fvalue associated with the corresponding transition from the ground state (0.23 [9]) is much too large to be identified with any observed absorption system in the C,H, spectrum. Adding a second diffuse species to the A0 basis with still smaller exponent (0 = 0.014) to represent the II = 4 member leads to a substantial lowering in the energy of this (second) mixed ‘(n,n*)-Rydberg state to 8.97 e\l, for ~!/practicalpurposes without altekg the stability of the lower-energy species discussed in section 5. Furthermore, the oscillator strength calculated for the corresponding transition shows nearly a fourfold decrease (to 0.07) compared to its artificial value in the more restrictive calculation with only one such diffuse Rydberg-type basis function*. Both the energy location and thefvalue associated with this second state in its proper representatiorz are in good agreement with Wilkinson’s experimental results for the R(Iv) bands in ethylene [14], w~hich are assigned in the original reference to a (ir,ndn) transition with possible pertrubatims from the lower-lying ‘(n,n*) r!aletzce state. Realization that a pure (?r,4dn) Rydberg transition should show anfvalue somewhat m the 0.005-0.01 range emphasizes that the state in question, despite its heavily_ diffkse conzpositiorz, still contains a significant amount of of valence (zJ*) c h aracter; the calculatedfvalue 0.07 would correspond to IO-15% valence state mhing in this case. Clearly this process of mixing the 1(.rr,.ir*> valence state with ‘(rr,ndx) Rydberg species will continue up
l
The prcscnce of the extra diffuse function does not change the transition energy to the hoest l(~,z*) state (8.1 eV), and it causa
the correspondingfvzlue
to decrease by only
0.006.
419
V&time 36, number
4
CHEMICAL PHYSICS LETFERS
1
Deceinber
1975
through higher quar: turn numbers. The j’value for the rid-d l(~,niL)--l (n,Jldn) mixed state in the calculation ,witt-two diffuse AO’s is s!iU too large (0.10-o. I5 [9j) tc be credibly identified with a [rue experimen-
in which it is not. three distinct cases can be distinguished according to the second key factor, namely the enera separation between the unperturbed (pure)
tal transition. Addition of more diffuse AO’s with stiU 2mzLler exponents to the basis set will clearly
In case I, in which the valence state lies well below all members of the various Rydberg series, one expects essentially no mixing effect (fig. I). Experimentally it is known [ 1 S] that Rydberg states possess term
Rydberg and valence levels.
result in a litrC/ler pmtitioni~tlg of‘rllis residrrai (n,n*) valehce4rell charac:er anmlg the higher-order mem hers of the ‘( z,ndz) Ryclbbcr,- mmuf~~d until cventually the iirnit of ionization is reached. Since the capaciti of a pure Rydberg state to mix in valence character should decrease I‘airly quickly as the size of the upper orbital increases (thereby causing the hamiltonian interactix mztri:; element with the pertinent valence state to become even smaller) it seems clear that after II Increases to fairly large values the deviations of the various Rydberg lines from their normal (unperturbed) pattern will all but disappear.
6. Mode! for mixing
of valence
The probability
of signilicant
valence
separate
and Rydberg
states
and Rydberg mixing
clcnrly
values
states
Ion
--
W, I n-_5 MCI
I
.’
426.,-
Case
* That is, the minimal energy splitting bctwcen upper end lower ,nixcd states xe found to vary from 0.0 to 1.0 eV. As discussed in section 5, vnlucs of MvRshould dccrcusc krly quicklv with the Rydberg quantum number II. ** There is some theorcticd support for the idea that H,R tvill bc largest
\vhen
H
ton
I’ i
Case Iii
,
m
I I
_----_-
PI<
differ
by only
ValFAlCe Ion, n-5
etc i
n=L m,red
i
I
Ryd-
krg
n=3 1
valencel
I
I I
t t
diagram showing the interaction
states
but in view of the results for the 3E; smtcs in O2 (section 2) it is clear that signiiicant cffccts can also occur when a double excitation separates the unperturbed slates.
---
the unperturbed
a single cscitation
I ! I I I ; , v z!cz!z~~{xK-=
n=3
IP) in the
I
I
n=L
-Valema
Fip. i. Schematic
I
rhe corresponding
on two
hamiltonian matrix element H,R connecting the VHlence species with a given Rydberg state. The calculitions for 0, and ethylene discussed above indicate that this quantity can lie anywhere in the range from 0.0 to 1.0 eV* for the lowest member of a Rydberg series. if HVR is quite small ‘very little mixing will occur, but for the (apparently numerous) instances
Case
below
between
depends
‘The first is the magnirudc of the
factors.
(energies
3.0-3.5 eV range for 3s, 2.2 to 2.8 eV for 3p and 1 .O to 1.8 eV for 3d members respectively. Thus if the maximum absorpticn in a given band system lies in the order of 7.0 eV below the expected location of the lowest-lyingpltrc Rydberg species of the same mixing is very unlikely. Good examples symmetry*, for this situation are supplied by the 3(n,r*) and the 3T1(n,+) valence siates in ethylene [l I] and formaldehyde [ 16,171 respectively. The minimum IP in ethylene is 10.51 eV and hence the lowest-lying Rydberg stare of the required symmetry in this instance, namely 3(7r,3d=), should be found no lower than roughly 8.5 eV. At an energ’ of 4.32 eV [I 81
between
valence and Rydbeg
states in
the
three cases discussed
in the text.
Volume 36, number 4
the 3 (z,fl*) valence trum
CHEMICAL
state is thus too low in the spec-
to be significantly
influenced
by the Rydberg
states of ethyiene. Similarly the 3~1(n,~*) states of HzCO found at 3.4 and 3.8 eV respectively are much too stable to interact noticeably with the pertinent 311(n,3dn) species (the IP for the n orbital is 10.86eV). Jn all these examples calculations including diffuse (Rydberg-type)
basis functions
very small tendency any degree
have shown
for these valence
of Rydberg
only
a
states to assume
character_
the situation in which the pertinent valence state roughly coincides in energy with the low-lying members of a Rydberg series of like symmetry (fig 1). Under these conditions the lowest adiabatic (CI) state can be fotind considerably MOW (by HVR) the expected location of the (pure) lowest-lying Rydberg state of this symmetry. In addition it is very likely that a good portion of the (unperturbed) valence character will be partitioned among the various higfler-order members of the same Rydberg series, as discussed in section 5. The often treated ‘(7,7-r*) absorption in C,H, and the 3Z, and 311,, - X transitions in O7 near 1.2 A appear to fall into this category* and numerous other examples which aiso fit this description are known, such as the ‘(FJ*) transitions in diimide and acetylene, for example [20,2 1 ] . The third and final case then involves the occurrence of a valence state which lies beyond tfle location of the minimum IP of the system (fig. 1). In this situation it is possible to have a large (usually infinite) number of Rydberg Sates lying below the valence species. The similarity with case I is obvious; if the value of HVR is quite large in a given instance some mixing between the Rydberg (especially the lower members) and the valence states is nevertheless possible. A good example for this behavior has been found in the study of the ethane spectrum. The location of the ‘(a,~‘) state in this system is found to vary depending on the number of diffuse (3pa) AO’s included in the A0 basis, thereby indicating that some interaction with the Rydberg states of like symCase 11 then
* The pertinent
involves
(unperturbed) 3(ng,3p) Rydberg species in O2 should have vertical excitation energies of 9.2-9.6 eV [5,6] while the pure ‘(;;,3dn) state in ethylene should tie in the 8.7 eV region [191,i.e. only1.OCV above the Frznck-Condon maximum in the V-N absorption system.
PHYSICS LIXTERS
1
December
1973
metry lying below the valence state is in fact occurring. This point is emphasized by the unusually largefvalue (0.15) calculated for the ‘(cr,3po) transition in ethane at 9.5 eV [23]. Because of the very high oscillator strength for the pure ‘(~,a”) valence-shell transition (probably 0.8 to 1 .O) even a small admixture of such valence character into a normal 3~0 Rydberg wavefunction can have a noticeabk effect on thef‘value of the corresponding transition to the resultant (weakly) mixed state. The same effect appears to be present to a somewhat lesser degree in the ‘(u,4po) transition, for which an fva!ue of 0.07 has been calculated (using gaussian exponents of 0.017 and 0.0!3 to represent the first two Rydberg orbitals). At the same time the calcs!ations indicate that the valence ’ (a,~*) state should be found at ro,ughly 16 eV with an /‘value of from 0.65 to 0.80, all of which results in turn agree reasonably well with the measured absorption and electron impact spectra of ethane [23,24], including the existence of a broad maximum in the continuum The indication is thus that even at 125000 cm-l. when a valence state is well separated from all Rydberg levels, a mixing effect can still be observed if bothH,, as well as the fvalue for the pertinent valence rransition are quite large (and the same would seem to hold in an analogous manner for case 1); on rhe other hand, from the standpoint of the spatial exterzsiorz of the corresponding upper orbital little or no indication of such a weak m&.ing is noticeable [22].
Acknowledgement
The authors wish to thank Professor C. Sandorfy and Dr. H. Lefebvre for various: stimulating discussions on the subject of Rydberg versus valence states and the interaction of corresponding potential surfaces. The services and computer time made available by the University of Bonn Computational Center are gatzklly acknowledged.
References [ 11 R. Pariser 2nd R.G. Parr, J. Chem. Phys. 21 (1953) 466, 767; J.A. Pople, Trans. Faraday Sot. 49 (1953) 137s. [2] W.C. Price, Phys. Rev. 47 (1935) 444; 5. Chem. Phys. 3 (1935) 256.
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PHYSICS LETTERS
[ 3 ] P.E. Phihipson and R.S. lviullikea, J. Chem. Phys. 28 (1958) 1248. (41 R.S. XIuU;:kcn, Inrern. I. Quantum Chcm. 5s (1971) 83. (51 R.J. Buenker and S.D. Peyerimhof’f, Chem. Phys. 8 (I 975) 324. (61 R.J. Buenker and S.D. Pcyerimhoff, Chem. Phys. Lctters 34 (1975) 225. (7 1 A.E. Douglas, in: Chemical spectroscopy and photochemistry in the vacmlm ultraviolet, eds. C. Sandorfv, PJ. Ausloos and X1-B. Robin (Reidel, Dordrecht. 1974). (B] RN. Huebner, R,J. Celottc, S.R. hficlczarek ;Ind C.E. Kuystt, Bull. Am. Phys. Sot. Nov. 1974, DCS, p. 1194. [9] ,R.J. Buenker and SD. Pcyerimhoff, Chem. Phys. 9 (1975) 75. [lo] A.J. Merer and R.S. hlulliken, Chem. Rev. 69 (1969) 639. [Ill R.J. Bucnkcr, S.D. Peyerimhoff and W.E. Kammcr, J. Chem. Phys. 55 (1971) 814. !12] E. hiiron, 9. Rx and _I. Jortner, J. Chem. Phys. 56 (1972) 5265. !131 M. Krauss and S.R. Mielczuek, J. Chem, Phys. 51
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1975).
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1141 P.G. Wilkinson, Can. J. Phys. 34 (1956) 643. [IS] C. Sandorfy. Lecture given at the VII International Conference of Photochemistry, Jerusalem, Israel, August 1973. [ 161 R.J. Buenker and SD. Peyerimhoff, J. Chem. Phys. 53 (1970) 1368. [ 171 J.L. Written and Xl. Hackmeyer, J. Chem. Phys. 51 (1969) 5584. [ 181 J.H. hloore Jr., J. Phys. Chem. 76 (1972) 1130. [19] U. Fischbach, R.J. Buenker and S.D. Peyerimhoff, Chem. Phys. 5 (1974) 265. (201 R. Vnsudevan, S.D. Peyerimhoff, R.J. Buenker and W.E. Kammer, Chem. Phys. 7 (1975) 187. 1211 W.E. Knmmcr, Chem. Phys. 5 (1974) 408. [22] R.J. Buenker and S.D. Peycrimhoff, Chem. Phys. 8 (1975) 56. [23] C. Sandorfy, in: Chemical spectroscopy and photochemistry in the vacuum ultraviolet, eds. C. Sandorfy, P-r_ Ausloos and M.B. Robin (Reidel, Dordrecht, 1974). [24] E.N. Ussettrc, A. Skerbelc and M-A. Dillon, J. Chem. Phys. 46 (1967) 4536;48 (1968) 539; 49 (1968) 2382.