- Email: [email protected]
Statistical analysis of photoelectron and Auger energy shifts in ionic solutions
and
,
Hans SIEGBAHN InstitutrC of Ph_wici, Unicen-ixyof Uppsala,Box 530. S-7-U 21 Uppda. Sweden Racked
2 September 1984
A previously ds&ed statistical model for sobation shifts of ionization cncrgks of atomic ions is cxtcndai to~Augcr transition cncqicx The model cxprcsscs the ionization sbifs the Auger cncrgy &_ft and the sum-of thy skifts in terms of cxpatation vaks taken over radial and an,@ar distriiution functions for the solvent Numerical cvakat~on is pcrfornkd for some monovaknt .anions and &ions and f&r Mg--+ in aqueous soh&xu_ Results from tin model are Compared pith cspcrimental data for absolute photockctron and Au&r shifts as W&I as for the individual wntriiution to sol&on cncrgics that can bc asxssed from a wmbm_& use of ESCA data for phokckctron and Auger cneqg shifts.
I_rrItroduction Core electron spectroscopy of liquids and solu-
Sons has now provided a large body of data on core level and Auger shifts for ionic species and pure liquids_ These shifts (between the free state and the Iiquid or dissolved state) are related to the potential at the site of the electronic transition and the dielectric response of the medium upon the transition_ ,The verticaI nature of photoekctron and Auger transitions implies that omy the highfrequency part of the dielectric response is represented in the shift. The potential and the dielectric response has ~-a functional dependence on. the medium structure and hence on temperature, cona&ration -and different solvent quantities related to statist@ thermodyna&s.~In the present contribution we evahtate a model which formulates the photoelectron and Auger shifts for ionic sohttions as statistical expectation values of the potential_ These expectation values invoIve the: radial ion-atomic and angular ion-dipoI& distribution functions as 0301-0104/85/$03.30 Q EIsevier Science Pub&hers
(North_Houa&
phyG=
P&&kg
DImsion)
obtained from statistical simulation techniques. A “hard” character of the ions is assumed fin that specific short-range effects are neglected_ .~. The vertical nature-of -the electronic transitions alIows identical distribution ~functions to be empIoyed for initialand final state of the transition in the sta&tical model_ Com+ring shifts with thermochemical salvation data this feature ako _ gives the possibility to deduce. a vahte. for the .dipoIe reorientationaI contributions to. soI+.ion This is achieved by proceeding .a Born-Haber cycle nhich equates the solvation shift of the photoelectron (or: Auger) transition with a difference in salvation energy between the initiakttate -ion and the hypothetical core-hole contain@g ion .(see fog_ 1) Replacing the core-hoie ion with its equiv- -_ aknt core .-ion the-:cycIe can be closed with the : salvation energy of the latter plus. the reorientational term_ This term emanates from thefact that. the sol&ion data of. the two ions are:. obtained ftim~.reIaxed surrou@ing stnk~~--.-~hiIcwhile:lthe 1strticture (or distribution: function) -mu.- g&me&&I -.L :- : .. : : ‘. : remains.frozen during the ioni+ion
; :
_
B-V.
: ? ----- ~~
_.
._
._
. -:
’
--
--r.z+----
~KeQo)T-
a
Z*l. diiar
reorientaticm
As shown in the present paper the statisticai model gives valuabIe information on the nature of the salvation shift for core photoehzctron and -Auger transitions and enables a partitioning of these shifts into physically well-defmed contributions The combined use of electron binding energy and Auger energy data and a parallel analysis with the statistical model is profitable in this respect_
2
s-
analysis
As for intermolecular binding energies there are two basic theoretical chemical approaches for the cahxriation of phase transition shifts (ET!%) of electronic bin&n= ener+s (BEG); the perturbational approach and the supermolecuhxr approachIn the perturbational approach one can partition the intermokdar contributions into physicaBy meaningful quantities, viz Coulomb, dispemion, induction and exchange energges and smaller mixing terms_ For the electronic binding energy PTS these quantities should be evaluated differentially between initial and f& states. which for general systems becomes an elaborate task to carry through For tie evaluation of the ETS in ionic sotutions. however. several simplifications may be undertaken, viz neglect of d%persios short-range exchange, couphng between initial and exterttitl induction effects and negIect of the charge transfer part of the induction energy These simplifications stem from the fact that the Coulomb and induction (polarization) parts of the PTS are an order of
magnitude larger than in neutral molecular liquids or solutions_ As discussed below they are. better motivated for cation than for anion soIutions_ Supermolecuhu and perturbational models are microscopical in the sense that they account for specific electronic interactions and solvent structure_ Dielectric models utilize on the other hand macroscopic data_ viz dielectric constants and salvation energies. in estimating the J?TS_ In its simplest form represented by the Born approximation [lj the dielectric model completely neggects the ordering of the soiven: around the dissolved ion_ E..tensions of dielectric models incorporating some considerations of specific structure have been proposed by Duke et al. [2,3] and applied to the ETS of solid organic compounds. Supermolecular cahxdations on the FTS of ionic solutions have been carried out by Arbman et al. and are reported in ret [a)_ The BE salvation shift originates from the potential at the site of ionization and from the external repolarization that follows from the change of the ionic the-e_ Both quantities evidently depend on solvent structure_ At the time for the (instantaneous) ionization this structure can be represented by various distribution functions. which, in some cases can be obtained from diffraction experiments or from statistical simulation techniques_ In the statistical model the BE shift is evaluated as an expectation value over radial atomic and dipole an$e distribution function g&RR) and X,(e). g,K(RK) gives the densitynormal&d probability of fmding the solvent atom K at a distance R, from the ion I and X,(B) gives the probability correlation between the solvent mokcuie dipole an&e and the ionic field for ion-mole&e pairs with a center-of-mass distance less than or equal to the radius of the first salvation shell_ Mathematical deftitions of these functions are giwzn in refs. [5;6]_ We introduce the basic pokization and Coulomb quantities:
H_ t&m
Ii. Sie&dm / Phomelertron an! Augerene& &iftxin ionic
.
where N~*=Xt,(O)d6=
(X,(iV)).
(3)
Here (X,(M)) is the expectation value of the coordination-number distribution function [6] in the first solvation shell, R,(B) the center of charge distance computed from the fiied mean value of the center of mass (cm.) distribution, and ito is the solvent molecule dipole moment. I&,,, u denotes a small rest term ‘.emanating from the Coulomb potential of the solvent-beyond the first solvation shell_ It is evaluated by-means of Fourier transformation technique involving the radial distribution functions, see ret [7]. The expression for (P) can be seen as the statistical expectation value of the Seneral pohuization expression
&!tihns
:
__.._1
:- -‘-j$.
and anioni as wellasfor the M~?..ion,indqueo~~. solution are-given- in table 1, and are there con- i pared with corresponding experimentalvalues for. glycol sohnion_ Analysis of these rest&is given in section4_ .. .. _. _. .. .~ -The binding- energy (BE) and At+& -knergy (AE)shiftsaredefinedas -. ::. E,,
= I&,-dv-
AhE
-
EGs, .-
E AE = &a+-AErp=
(5) ..
EB*E- EBy
(6) -<+, :
E,od+‘y
E_&-
E,g;s_
(8)
Letting Qt denote the charge of the ion (cation, anion or atom) in its ground state one finds (initial and final state quantities separated by brackets): AEs=
[(Q1 + l)(V)
-(Q,
+ l)‘(P)]
?.a
where a spherical average, PK, of the atomically decomposed [S] solvent polarizability tensors (A,) has been undertaken (E(‘) is here the electric field at the solvent position LI” excerted at io&ation site ONumerical resuits for BE and AE shifts and for (P) and (V) values for some monovalent cations
A%?
- [QlV>
- QI’(P>l
= - (32,
+- w;,
= [@I
+ 1)0’>
- [
+(2Q,
-
(9)
tQ, + h’>] - (QI + ?-l’(P)]
+3)(P).
(If0
Table 1 Tbeoreticd
and cx~htal photodcctron biding cncz=y and Auger cktron eneqy shifts (CL’). Cchxms l-5 display results from whdxions with the statistical model for aqueous solutions and columns 6 and 7 display eqetimentat rest&s for glyco~ sohtic~n Calcuhtions USCdistribution functions from the zMonteCarlo simuhions reported in ref_ [9] for the monovalent ions and iiom the mokuhr dynamics simulations reported in ref_ [IO] for Mg’*
AEP”‘”
AEgg--
dEgy
Li+
-
Na* K’ F-
-4.06 --La i-262
3-77 L79 1.94 3.07
a-
tzio
MS”
I_85
-781
3.80
-1132 -06 - 583 -I-3-07 i-185 - 19.00
- 16-32 -12X?tD-854 -f-5.69 f455 -2681
-153 - 124 - 9.8 +99 t 5.6 -23.8
5.00
AAygy + 1737 -!-1241 + 0.45 - 0x5 + 34.42
Expectation vale for Chlomb energy. cq_ (2)_ Expasation value for pohriation cttagy. cq_ (1). Polatization amtriiutiox~ to BE shift Photodaztroo binding stagy shift. a& (9) (water)_ hkasurcd pt~otoctccmx~ binding atergy shift (gly.col)_ Au& dcctron bmding axzgy &if< q_ (10) (water)_ @ bfcasurcd Auger binding energy shift (&wl)_-” Invase hctivc iotkradius (au).
A
IT=- y
+ 19.1 -22
l/C&h’ 0.76 055 O-40 O-43 0.28 O-79
4 ‘I d d) =’ 0
.._
.
This AEE
gives the f AEs
well-known =2(P).
Formula
equalities (11)
For all Q,_ This derivation negkcts saturation eFFects between various degg of ionization as u-e11 as any coupling between the internal ionic rekutadon and the external polarization- Using radial distribution Functions from Monte Carlo simulation (Na+, [9B and mokcular dynamics simulation (Mg’*_ [lOl, data we obtain (P) values for Na’
and Mg” in aqueous solution of 28 and 3-S eV respectivety_ This can be compared with the vah,te
of 335 eV obtained From measured BE and AE shifts For Na’ in glycol so!ution [1X]_
3. ianalysis ofi ESCA
data
Chemical shifts in free molecules and solids have previously been analyzed by means OF Born-Haber cycles invoking thermochemical heat of Formation (solvation) data and the 2 + L equivalent cores approximation This approach For salvation shirts in liquids and soIutions involving combined binding and Auger energg data was developed in reF_ ill]_ An important particularity of this scheme For iiquids or solution results from the stmctural reorientation of the medium between the initial and final states of the ionization_ A Born-Haber cycle for core ionization in ionic solutions is displayed in fig lStarting out From the bare ion (here K’) in solution au alternative path to the corresponding iouized species is given by Fiit dissolving the ion Followed by ionization and solution of the ionized species The last step is hypothetical in the sense that the lifetime of a core-hole species is too short For a nuclear rekation to occur_ Adopting the 2 -t- 1 approximation the cycle can akmatively be closed by the sokation energy of the corresponding ion of one higher ck-e unit (here Ca’*)_ Since photoionization is vertical with only tlte high-Frequency part of the dielectric response induded, the last cycle leaves out a quantity referring to the change in equiliirium stmctmz of the K* and Ca” ions The cycks are expressed by the
EE(K*) --E%K+) = E,,[K’(2p)]
-E&K’).
E&G’*)
A-_=
-
(12) E,,[K+(Zp)]_
(13)
The reorientational term contains both a direct change in the potential and a change in the electronic repolarization due the altered structure_ It
amounts
to a few eV for monovalent-ions
and
follows a r&ar trend with respect to the effective ionic radius [ll]_ rlE_y can be evaluated From a Born-Haber cycle. see fig 1, that is fully analogous to the one for BEs, but with K*(2p) replaced by K”(2p’) and with a dipolar reorientational term obtained from the solvation energy of Sc3* ion [cF_ eqs- (12) and (1311. Using eqs- (12) and (13) the solvation shift for a general ion can be partitioned into one orbital part and one relaxation part AE’“” BE
= AEdW - AEz_
(14)
in Iine with ordinary theory For the chemical shift. These two parts can be identified by means of a combined measurement of AE and BE shifts:
-!-AEB~=2AE,?$!&
05)
In the statistical model cq_ (15) equaIs 2(P)_ The orbital energy shift is a result of the static and induced-solvent potential_ Assuming spherically averaged polarizabilities the statistical expectation value of the latter quantity (dipole potential) can be evaluated as
AE~=~((LC~K
= 2bE2,
R,)/R:,) =C
06)
where ux and Ed denote induced-dipole moment and the ionic lield respectively_ Thus the induceddipole potential contribution to the orbital energy shift doubles the polarization Free energy of the initial state This relation_ can be derived From
ordinary electrostatic theory [12]. For the static part of the orbital energy shift we then find: -
= Ae-
- Q;(P)_
07)
Within this decomposition scheme the total solvation energy of the ion can be written as Ed,
= AE,,
+ Ae_,
- E-,
(18)
is the solvent-solvent interaction where ES&, (strain) induced by the salvation of the ion_
4, Comparison ESCA data
between the statistical model and
From the above it is clear that the ESCA analysis allows the different contributions to the solvation energy to be decomposed_ This rests on the simultaneous measurements of Auger and photoelectron binding energy shifts and the assumption that saturation effects between first and second ionization steps can be neggected [expressed in eqs_ (15)-(18)]. In the perturbational model, described in section 2, the Coulomb and relaxation components can be directly evaluated by the calculation of the (P) (cf. AESI) and the (V) (cf. ArSUr) quantities_ The accuracy of this approach evidently depends on the approximation undertaken, as described in section 2, and of the quality of the employed distribution functions_ We apply the present analysis to the sodium ion_ Basic theoretical and experimental values are found in table l_ The use of eq_ (15) gives a relaxation contribution to the glycol solvation energy of 3.35 eV, which leads to an induced potential contribution to the orbital energy shift of 6.70 eV, cf_ eq. (16) This in turn leads to a static contribution to Ae of 235 eV, cf_ eq_ (17) Comparing with the experimental solvation energy of 39 eV [13] one fmds a strain energy of 1.8 eV. The statistical model calculation for the sodium ion in aqueous solution gives a relaxation energy of 2-79 eV and a static potential contribution of 4-02 eV_ This gives an induced potential contribution to the orbital enshift of 558 eV_ Comparing with
the solvation energy of 3.91 eV for sodium i0tn.m water [13], eq. (18) Bredicts a strain v&e of%&:‘ .: .__ in&is-~ -. Although the theore&aI and &&rimental data. are ob:ained for different solvents they a&of the same magnitude and follow similar trends as can . be seen in table I_ Thus de observation of a~line5r relationship between the BE shift and inverse ionic radius is theoretically confirmed.‘ Only when the stat?t;%tical nature of the solvent structure is considered can this relationship be accounted for. Most of the differences in the BE shifts for the=various ions can be allocated to the first solvation shell, where the solvent structure is highly ordered, whereas beyond &is region the weak structure gives (V) and (P) contributions of similar m&gnitudes for the different ions. The polarization and Coulomb contributions work cooperatively for the BE solvation shifts, producing negative shifts for the cations and positive for the anions. Concerning the AE shifts one finds that the polarization and Coulomb contributions add up to large positive shifts for the~cation.., while for the anion shifts the two contribtttions may or may not work cooperatively depending in this case on the ground state charge_ In contrast to the BE shifts the AE shifts will differ substantially in magnitude betweeu anions and cations of- the same absolute charge, cf. table l_ The agreement with experiment for the AE shift is satisfactory for the Na’ ion considering that the values are bbtamed for different solvents_ The shifts for the fluorine ion constitute an exception from the theoretical (water) and experimental (glycol) agreement. It is plausible that short-range effects are more important for anions than for cations. Fluorine forms a very short hydrogen bond with the solvent, which may be of consequence for the nc@ct of exchange and charge transfer interactions Also the approximation given by spherically averaging of the atomically decomposed polarizab&ties is more crucial in this case_ The cumuIative radial polarization contributions to BE shift for Na* in aqueous solution are given in fig. 2_ The effect of the first solvation shell edge c&r clearly be discerned in the plot. More than one third of the total electronic polarization energy is
investigationk as a prerequisite A major advaktge with a perturbational approach to solvatioti shifts is that it lends itself to a statistical formulation as has been proposed in the present work_ This is criticaI for the analyGs of the BE and AE shifts alone, but makes aIso possible an extension of this analysis to deal with effects on the shifts of tcmperature and concentration variations in the solvent and to various broadening mechanisms of the solution energg levels_
References
seen to originate from the reggon outside the first solvatiou shell_ The present perturbational approach aIIows for an identification of the Coulomb and polarization contributions to the BE and AE soIvation shiftsLong-range effects can be incorporated and special duster effec*s avoided_ Short-range effects, such as charge transfer and exchange which may he of increased importance for e-g_ soft ionic sohxtions, and aIso coupled interna&external reIaxation can in principle be incorporated in the present approach but would caII for extensive ab initio
[I) M_z Born. z_ Pb>* I (1920) 45. 121CB_ Duke 1V.R Salanek TJ. Fabish_ JJ_ Rifsko. HI2 Tbomw and A_ Paton Php Rev_ B18 (1978) 5717. [3J C-R Duke in: Emxukd linear chain compoupds VoL 2. cd_ J_S_Miller (Pbsum Rar Xeu- York 1982) p_ 5R [4J M A&man. H_ Skgbabm I_ Pcmxsson and P_ Skgbahn. Mole Pb,r IO be publisbcd. !SJ ?-% E+staff. An introduction to the EiqGd Marc
Pew_
x3-
Yctk
1967).
f6J .M_ Maei and DJ- Bemidge J. Cbcm Phys 76 (1982) 593. [7J H _&en and H_ !&&mhn. J. 5 Php SI(198-4) 488. 181G_ Karlstr&n. Tbcurct C&m Acta 60 (1982) 535. (9J M_ Me& and D-L Bemidge. J. Cbcm Ph>s 74 (1981). [IO] W_ Dick 0. Ricdc and K_ Hchzingcr. 2 ~amforsch 37a (1982) 1038_ [ll] H_ Stcgbahn. M_ LundhoIm. S- Holmbetgand hf. Arbman. Phpica Saipfa 27 (1983) 431. (IZ] C_i_F_ meur_ Theozy Of ezlemxk &3oiaeaLion VOL 1 (Elsmicr. Asisferd;ul; 1973)_ 114 R CZimxr and G. Tryor~ J. Chum Ph>x 66 0977) 4413s
,
Hans SIEGBAHN InstitutrC of Ph_wici, Unicen-ixyof Uppsala,Box 530. S-7-U 21 Uppda. Sweden Racked
2 September 1984
A previously ds&ed statistical model for sobation shifts of ionization cncrgks of atomic ions is cxtcndai to~Augcr transition cncqicx The model cxprcsscs the ionization sbifs the Auger cncrgy &_ft and the sum-of thy skifts in terms of cxpatation vaks taken over radial and an,@ar distriiution functions for the solvent Numerical cvakat~on is pcrfornkd for some monovaknt .anions and &ions and f&r Mg--+ in aqueous soh&xu_ Results from tin model are Compared pith cspcrimental data for absolute photockctron and Au&r shifts as W&I as for the individual wntriiution to sol&on cncrgics that can bc asxssed from a wmbm_& use of ESCA data for phokckctron and Auger cneqg shifts.
I_rrItroduction Core electron spectroscopy of liquids and solu-
Sons has now provided a large body of data on core level and Auger shifts for ionic species and pure liquids_ These shifts (between the free state and the Iiquid or dissolved state) are related to the potential at the site of the electronic transition and the dielectric response of the medium upon the transition_ ,The verticaI nature of photoekctron and Auger transitions implies that omy the highfrequency part of the dielectric response is represented in the shift. The potential and the dielectric response has ~-a functional dependence on. the medium structure and hence on temperature, cona&ration -and different solvent quantities related to statist@ thermodyna&s.~In the present contribution we evahtate a model which formulates the photoelectron and Auger shifts for ionic sohttions as statistical expectation values of the potential
(North_Houa&
phyG=
P&&kg
DImsion)
obtained from statistical simulation techniques. A “hard” character of the ions is assumed fin that specific short-range effects are neglected_ .~. The vertical nature-of -the electronic transitions alIows identical distribution ~functions to be empIoyed for initialand final state of the transition in the sta&tical model_ Com+ring shifts with thermochemical salvation data this feature ako _ gives the possibility to deduce. a vahte. for the .dipoIe reorientationaI contributions to. soI+.ion This is achieved by proceeding .a Born-Haber cycle nhich equates the solvation shift of the photoelectron (or: Auger) transition with a difference in salvation energy between the initiakttate -ion and the hypothetical core-hole contain@g ion .(see fog_ 1) Replacing the core-hoie ion with its equiv- -_ aknt core .-ion the-:cycIe can be closed with the : salvation energy of the latter plus. the reorientational term_ This term emanates from thefact that. the sol&ion data of. the two ions are:. obtained ftim~.reIaxed surrou@ing stnk~~--.-~hiIcwhile:lthe 1strticture (or distribution: function) -mu.- g&me&&I -.L :- : .. : : ‘. : remains.frozen during the ioni+ion
; :
_
B-V.
: ? ----- ~~
_.
._
._
. -:
’
--
--r.z+----
~KeQo)T-
a
Z*l. diiar
reorientaticm
As shown in the present paper the statisticai model gives valuabIe information on the nature of the salvation shift for core photoehzctron and -Auger transitions and enables a partitioning of these shifts into physically well-defmed contributions The combined use of electron binding energy and Auger energy data and a parallel analysis with the statistical model is profitable in this respect_
2
s-
analysis
As for intermolecular binding energies there are two basic theoretical chemical approaches for the cahxriation of phase transition shifts (ET!%) of electronic bin&n= ener+s (BEG); the perturbational approach and the supermolecuhxr approachIn the perturbational approach one can partition the intermokdar contributions into physicaBy meaningful quantities, viz Coulomb, dispemion, induction and exchange energges and smaller mixing terms_ For the electronic binding energy PTS these quantities should be evaluated differentially between initial and f& states. which for general systems becomes an elaborate task to carry through For tie evaluation of the ETS in ionic sotutions. however. several simplifications may be undertaken, viz neglect of d%persios short-range exchange, couphng between initial and exterttitl induction effects and negIect of the charge transfer part of the induction energy These simplifications stem from the fact that the Coulomb and induction (polarization) parts of the PTS are an order of
magnitude larger than in neutral molecular liquids or solutions_ As discussed below they are. better motivated for cation than for anion soIutions_ Supermolecuhu and perturbational models are microscopical in the sense that they account for specific electronic interactions and solvent structure_ Dielectric models utilize on the other hand macroscopic data_ viz dielectric constants and salvation energies. in estimating the J?TS_ In its simplest form represented by the Born approximation [lj the dielectric model completely neggects the ordering of the soiven: around the dissolved ion_ E..tensions of dielectric models incorporating some considerations of specific structure have been proposed by Duke et al. [2,3] and applied to the ETS of solid organic compounds. Supermolecular cahxdations on the FTS of ionic solutions have been carried out by Arbman et al. and are reported in ret [a)_ The BE salvation shift originates from the potential at the site of ionization and from the external repolarization that follows from the change of the ionic the-e_ Both quantities evidently depend on solvent structure_ At the time for the (instantaneous) ionization this structure can be represented by various distribution functions. which, in some cases can be obtained from diffraction experiments or from statistical simulation techniques_ In the statistical model the BE shift is evaluated as an expectation value over radial atomic and dipole an$e distribution function g&RR) and X,(e). g,K(RK) gives the densitynormal&d probability of fmding the solvent atom K at a distance R, from the ion I and X,(B) gives the probability correlation between the solvent mokcuie dipole an&e and the ionic field for ion-mole&e pairs with a center-of-mass distance less than or equal to the radius of the first salvation shell_ Mathematical deftitions of these functions are giwzn in refs. [5;6]_ We introduce the basic pokization and Coulomb quantities:
H_ t&m
Ii. Sie&dm / Phomelertron an! Augerene& &iftxin ionic
.
where N~*=Xt,(O)d6=
(X,(iV)).
(3)
Here (X,(M)) is the expectation value of the coordination-number distribution function [6] in the first solvation shell, R,(B) the center of charge distance computed from the fiied mean value of the center of mass (cm.) distribution, and ito is the solvent molecule dipole moment. I&,,, u denotes a small rest term ‘.emanating from the Coulomb potential of the solvent-beyond the first solvation shell_ It is evaluated by-means of Fourier transformation technique involving the radial distribution functions, see ret [7]. The expression for (P) can be seen as the statistical expectation value of the Seneral pohuization expression
&!tihns
:
__.._1
:- -‘-j$.
and anioni as wellasfor the M~?..ion,indqueo~~. solution are-given- in table 1, and are there con- i pared with corresponding experimentalvalues for. glycol sohnion_ Analysis of these rest&is given in section4_ .. .. _. _. .. .~ -The binding- energy (BE) and At+& -knergy (AE)shiftsaredefinedas -. ::. E,,
= I&,-dv-
AhE
-
EGs, .-
E AE = &a+-AErp=
(5) ..
EB*E- EBy
(6) -<+, :
E,od+‘y
E_&-
E,g;s_
(8)
Letting Qt denote the charge of the ion (cation, anion or atom) in its ground state one finds (initial and final state quantities separated by brackets): AEs=
[(Q1 + l)(V)
-(Q,
+ l)‘(P)]
?.a
where a spherical average, PK, of the atomically decomposed [S] solvent polarizability tensors (A,) has been undertaken (E(‘) is here the electric field at the solvent position LI” excerted at io&ation site ONumerical resuits for BE and AE shifts and for (P) and (V) values for some monovalent cations
A%?
- [QlV>
- QI’(P>l
=
+- w;,
= [@I
+ 1)0’>
- [
+(2Q,
-
(9)
tQ, + h’>] - (QI + ?-l’(P)]
+3)(P).
(If0
Table 1 Tbeoreticd
and cx~htal photodcctron biding cncz=y and Auger cktron eneqy shifts (CL’). Cchxms l-5 display results from whdxions with the statistical model for aqueous solutions and columns 6 and 7 display eqetimentat rest&s for glyco~ sohtic~n Calcuhtions USCdistribution functions from the zMonteCarlo simuhions reported in ref_ [9] for the monovalent ions and iiom the mokuhr dynamics simulations reported in ref_ [IO] for Mg’*
AEP”‘”
AEgg--
dEgy
Li+
-
Na* K’ F-
-4.06 --La i-262
3-77 L79 1.94 3.07
a-
tzio
MS”
I_85
-781
3.80
-1132 -06 - 583 -I-3-07 i-185 - 19.00
- 16-32 -12X?tD-854 -f-5.69 f455 -2681
-153 - 124 - 9.8 +99 t 5.6 -23.8
5.00
AAygy + 1737 -!-1241 + 0.45 - 0x5 + 34.42
Expectation vale for Chlomb energy. cq_ (2)_ Expasation value for pohriation cttagy. cq_ (1). Polatization amtriiutiox~ to BE shift Photodaztroo binding stagy shift. a& (9) (water)_ hkasurcd pt~otoctccmx~ binding atergy shift (gly.col)_ Au& dcctron bmding axzgy &if< q_ (10) (water)_ @ bfcasurcd Auger binding energy shift (&wl)_-” Invase hctivc iotkradius (au).
A
IT=- y
+ 19.1 -22
l/C&h’ 0.76 055 O-40 O-43 0.28 O-79
4 ‘I d d) =’ 0
.._
.
This AEE
gives the f AEs
well-known =2(P).
Formula
equalities (11)
For all Q,_ This derivation negkcts saturation eFFects between various degg of ionization as u-e11 as any coupling between the internal ionic rekutadon and the external polarization- Using radial distribution Functions from Monte Carlo simulation (Na+, [9B and mokcular dynamics simulation (Mg’*_ [lOl, data we obtain (P) values for Na’
and Mg” in aqueous solution of 28 and 3-S eV respectivety_ This can be compared with the vah,te
of 335 eV obtained From measured BE and AE shifts For Na’ in glycol so!ution [1X]_
3. ianalysis ofi ESCA
data
Chemical shifts in free molecules and solids have previously been analyzed by means OF Born-Haber cycles invoking thermochemical heat of Formation (solvation) data and the 2 + L equivalent cores approximation This approach For salvation shirts in liquids and soIutions involving combined binding and Auger energg data was developed in reF_ ill]_ An important particularity of this scheme For iiquids or solution results from the stmctural reorientation of the medium between the initial and final states of the ionization_ A Born-Haber cycle for core ionization in ionic solutions is displayed in fig lStarting out From the bare ion (here K’) in solution au alternative path to the corresponding iouized species is given by Fiit dissolving the ion Followed by ionization and solution of the ionized species The last step is hypothetical in the sense that the lifetime of a core-hole species is too short For a nuclear rekation to occur_ Adopting the 2 -t- 1 approximation the cycle can akmatively be closed by the sokation energy of the corresponding ion of one higher ck-e unit (here Ca’*)_ Since photoionization is vertical with only tlte high-Frequency part of the dielectric response induded, the last cycle leaves out a quantity referring to the change in equiliirium stmctmz of the K* and Ca” ions The cycks are expressed by the
EE(K*) --E%K+) = E,,[K’(2p)]
-E&K’).
E&G’*)
A-_=
-
(12) E,,[K+(Zp)]_
(13)
The reorientational term contains both a direct change in the potential and a change in the electronic repolarization due the altered structure_ It
amounts
to a few eV for monovalent-ions
and
follows a r&ar trend with respect to the effective ionic radius [ll]_ rlE_y can be evaluated From a Born-Haber cycle. see fig 1, that is fully analogous to the one for BEs, but with K*(2p) replaced by K”(2p’) and with a dipolar reorientational term obtained from the solvation energy of Sc3* ion [cF_ eqs- (12) and (1311. Using eqs- (12) and (13) the solvation shift for a general ion can be partitioned into one orbital part and one relaxation part AE’“” BE
= AEdW - AEz_
(14)
in Iine with ordinary theory For the chemical shift. These two parts can be identified by means of a combined measurement of AE and BE shifts:
-!-AEB~=2AE,?$!&
05)
In the statistical model cq_ (15) equaIs 2(P)_ The orbital energy shift is a result of the static and induced-solvent potential_ Assuming spherically averaged polarizabilities the statistical expectation value of the latter quantity (dipole potential) can be evaluated as
AE~=~((LC~K
= 2bE2,
R,)/R:,) =C
06)
where ux and Ed denote induced-dipole moment and the ionic lield respectively_ Thus the induceddipole potential contribution to the orbital energy shift doubles the polarization Free energy of the initial state This relation_ can be derived From
ordinary electrostatic theory [12]. For the static part of the orbital energy shift we then find: -
= Ae-
- Q;(P)_
07)
Within this decomposition scheme the total solvation energy of the ion can be written as Ed,
= AE,,
+ Ae_,
- E-,
(18)
is the solvent-solvent interaction where ES&, (strain) induced by the salvation of the ion_
4, Comparison ESCA data
between the statistical model and
From the above it is clear that the ESCA analysis allows the different contributions to the solvation energy to be decomposed_ This rests on the simultaneous measurements of Auger and photoelectron binding energy shifts and the assumption that saturation effects between first and second ionization steps can be neggected [expressed in eqs_ (15)-(18)]. In the perturbational model, described in section 2, the Coulomb and relaxation components can be directly evaluated by the calculation of the (P) (cf. AESI) and the (V) (cf. ArSUr) quantities_ The accuracy of this approach evidently depends on the approximation undertaken, as described in section 2, and of the quality of the employed distribution functions_ We apply the present analysis to the sodium ion_ Basic theoretical and experimental values are found in table l_ The use of eq_ (15) gives a relaxation contribution to the glycol solvation energy of 3.35 eV, which leads to an induced potential contribution to the orbital energy shift of 6.70 eV, cf_ eq. (16) This in turn leads to a static contribution to Ae of 235 eV, cf_ eq_ (17) Comparing with the experimental solvation energy of 39 eV [13] one fmds a strain energy of 1.8 eV. The statistical model calculation for the sodium ion in aqueous solution gives a relaxation energy of 2-79 eV and a static potential contribution of 4-02 eV_ This gives an induced potential contribution to the orbital enshift of 558 eV_ Comparing with
the solvation energy of 3.91 eV for sodium i0tn.m water [13], eq. (18) Bredicts a strain v&e of%&:‘ .: .__ in&is-~ -. Although the theore&aI and &&rimental data. are ob:ained for different solvents they a&of the same magnitude and follow similar trends as can . be seen in table I_ Thus de observation of a~line5r relationship between the BE shift and inverse ionic radius is theoretically confirmed.‘ Only when the stat?t;%tical nature of the solvent structure is considered can this relationship be accounted for. Most of the differences in the BE shifts for the=various ions can be allocated to the first solvation shell, where the solvent structure is highly ordered, whereas beyond &is region the weak structure gives (V) and (P) contributions of similar m&gnitudes for the different ions. The polarization and Coulomb contributions work cooperatively for the BE solvation shifts, producing negative shifts for the cations and positive for the anions. Concerning the AE shifts one finds that the polarization and Coulomb contributions add up to large positive shifts for the~cation.., while for the anion shifts the two contribtttions may or may not work cooperatively depending in this case on the ground state charge_ In contrast to the BE shifts the AE shifts will differ substantially in magnitude betweeu anions and cations of- the same absolute charge, cf. table l_ The agreement with experiment for the AE shift is satisfactory for the Na’ ion considering that the values are bbtamed for different solvents_ The shifts for the fluorine ion constitute an exception from the theoretical (water) and experimental (glycol) agreement. It is plausible that short-range effects are more important for anions than for cations. Fluorine forms a very short hydrogen bond with the solvent, which may be of consequence for the nc@ct of exchange and charge transfer interactions Also the approximation given by spherically averaging of the atomically decomposed polarizab&ties is more crucial in this case_ The cumuIative radial polarization contributions to BE shift for Na* in aqueous solution are given in fig. 2_ The effect of the first solvation shell edge c&r clearly be discerned in the plot. More than one third of the total electronic polarization energy is
investigationk as a prerequisite A major advaktge with a perturbational approach to solvatioti shifts is that it lends itself to a statistical formulation as has been proposed in the present work_ This is criticaI for the analyGs of the BE and AE shifts alone, but makes aIso possible an extension of this analysis to deal with effects on the shifts of tcmperature and concentration variations in the solvent and to various broadening mechanisms of the solution energg levels_
References
seen to originate from the reggon outside the first solvatiou shell_ The present perturbational approach aIIows for an identification of the Coulomb and polarization contributions to the BE and AE soIvation shiftsLong-range effects can be incorporated and special duster effec*s avoided_ Short-range effects, such as charge transfer and exchange which may he of increased importance for e-g_ soft ionic sohxtions, and aIso coupled interna&external reIaxation can in principle be incorporated in the present approach but would caII for extensive ab initio
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