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On the free energy changes of a solution in light absorption or emission processes
Volume 99, number 1
CHEMICAL
PHYSICS LEl-l-ERS
22 July 1983
ON THE FREE ENERGY CHANGES OF A SOLUTION IN LIGHT ABSORFTiON
OR EMISSION PROCESSES
R. BONACCORSI Istituto di Cldmim Quantisticned Energetica Molecolare de1 Ch’R. Via Risorginento 35.56100 Piss. Italy and
R. CIMIRAGLIA and J. TOMASI Zstituto di CJdmicaFisicn. Universirridi Pisa. 1% Risorgimento 35.56100 Pisa. Italy
Received 29 March 1983;in
final form 21 hiay 1983
A simple model to evaluate energy changes in dilute solutions produced by absorption or emission of light is proposed_ The computational scheme exploits some characteristics of the continuum model of liquids and is inserted into ab initio quantum-mechanical programs for isolated molecules. The model is addressed to evaluate changes in thermodynamic properties of the whale solution - while earlier models often consider the solute only -and it presentsa detailed decomposition of the phenomenon into successive steps.
1. Introduction
The processes of absorption and emission of light by a molecular solute in a solution cause several dynamical phenomena whose theoretical description requires a considerable effort. And the interpretation of solvent shifts on electronic spectra, though “prima facie” appropriate to energy transitions only, is confronted with these dynamical processes. The objective of getting manageable formulas of practical use has suggested drastic simplifications (for a detailed review see, e.g. ref. [l]). As far as the dynamics of the solute M is considered, the process is generally outlined by dividing it into separate steps, each treated in a time-independent formalism_ Attention is focused on vertical absorption from the ground state (GS) to an excited state (EX), at the GS equilibrium geometry, and on the emission from EX, in principle at a different geometry,selected on the basis of empirical considerations on the shape of the EX potential energy surface [2] _The dynamics of the rearrangement of the solvent molecules S is dealt with 0 009-26 14/83/0000-OOOO/% 03-00 0 1983 North-Holland
statistically (except for the cases in which it is necessary to introduce specifc interactions between solvent and solute, as occurs in the formation of excimers and in other intermolecular excitation transfers). Most methods make use of the continuum description of the solvent; in these models the dynamics of the solvent is represented by the changes in the reaction field due to the absorption (or emission) process, introducing a distinction between polarization effects due to the orientation of the solvent molecules (related to the macroscopic dielectric constant e) and polarization effects due to the induction of displacements in the electronic charge distribution of the solvent molecules (related to the square of the macroscopic refractive index n*)_ Recently we proposed a computational scheme [3] which translates the abovementioned features into the ab initio procedures of common use for gas-phase molecular calculations. In that paper (herafter called I) attention was focused on the evaluation of absorption and emission wave numbers at a given geometry of M, disregarding the energetic questions related to the 77
\rolumc 99, number 1
22 July 1983
CHEIIICAL PHYSICS LETTERS
other dynamical aspects, summarized above. We give here a complement to that scheme in which energy changes of the whole solution during the light absorption/emission processes are considered.
2. Theoretical aspects
01
=
qot(PGs4 -
(2)
3
In eq. (2) we have indicated the parameters on which (Tdepends, namely the GS charge distribution and the static dielectric constant E. We remark in addition that: oto&GS?
E) = o,,(PGS)
+ %&GS-
lz7) -
(3
The method referred to in paper I. which we shall not summarize here. was concerned with the application to the special case of the electronic band shift of a more general procedure [3] to deal with solvent effects_ It will be sufficient for the present purposes to recall that the solute-solvent interactions are introduced in the molecu@r hamiltonian by means of an additional operator I’, expressed in terms of a polarization surface charge distribution u accommodated on the cavity surface:
The total and inductive components of u can be computed separately, while the orientational component is defined as a difference(2) M absorbs a photon of the correct energy and the solute passes from M,, to ME,. The solvent esperiences the changes in the M charge distribution through its inductive part only, because the electronic transition time is muc11 shorter than the orientational relaxation time. The polarization vector is now Pz. related to
f3,t = @, + I;b _
02 = o,,@&
(1)
The charge distribution u is a function of the solute cl~irge distribution p5t (including both electron and nuclear contributions) and is due. according to the case. to the orientatiomd. inductive or total component of the polarization vector Pof the dielectric continuum simuking the solvent. The method of calculation of u depends on the spectroscopic process under consideration_ When the appropridte u. and then ?a_ is obtained. the difference in energy between the two states involved in the electronic transition is computed with current ab initio procedures. The ewluation of energy changes in the solution
produced by light absorption or emission requires some rearrangements of the computational scheme outlined above. JS we shall elucidate in the following schematic model example. Let us consider a dilute solution. for which the customary assumptions arc accepted (sufficient separation of the charge distributionsof Al and S, electronic excitation confined to hi. lack of dielectric saturation phenomena. etc.) and for which also the change in internal geometry of hl is negligible_ We consider now a simple cycle which passes through the following states: (1) The solute M in the GS is in equilibrium with the solvent_ The charge distribution pu gives rise to a polariration P, of the dielectric. accounted for by a surface charge distribution: 7s
+ o,,(,+Y-
n3-) -
(9
u, is introduced. via the corresponding operator, into the hamiltonian, and the absorption frequency is calculated using this hamiltonian. (3) hlEx has a sufficiently long lifetime to rearrange completely the solvent around itself. The solvent reaches a new polarization state P3.related to 03 = U,,&JEX. E) -
(5)
(4) Through an emission process (non-radiative decaysare neglected here) the CS is reached again. During this process only the inductive part of u is changed, and the calculation of the electronic level difference is performed by using a pa operator. corresponding to a polarization state P4 related to u4 = uor(PEX) + uind @GS 71z2) -
(6)
(5) Equilibrium of the solvent occurs and the same physical situation as in state 1 is reached. The evaluation of the electronic transition wavenumbers corresponding to steps 1 -+ 2 and 3 -+ 4 have been performed in paper 1 by considering the solvent as a bath, thus disregarding other energetic changes. The energy change in the whole cycle may be described with the diagram given in fig_ 1_ This diagram refers to a conventional space having as coordinates the energy and the state of polarization of the dielectric_ This last coordinate has not in the drawing a quantitative definition (even though when necessary a
Volume 99, number 1
CHEMICAL PHYSICS LETTERS
22 iuly 1983
which gives the difference of W with respect to a state composed of gaseous M (with the charge distribution &) and the separate dielectric containiig the appropriate cavity, assuming moreover that phf = & (i.e. disregarding polarization of M in the dielectric) [9,10] _ Returning now to our model, it is evident that a quantity analogous to eq. (8) and related to state 1 of our cycle is
ClO) F&_ 1. Schematic relationship of the four steps considered in the model cycle in s solvent pohrizationenergy coordinate frame. 1 refers to hfGs in equilibrium with the solvent; 2 to h&x with solvent not in equilibrium;3 to MEXin equilibrium with the solvent;4 to MGS with solvent not yet equilibrated.
Steps I--, 2 and 3 --+4 respectivelyinvolveabsorptionand emission of li$t.
more realistic scale could be elaborated). For a proper de~mit~ouof the energy it is convenient to consider the corresponding classical case_ Application of classical electrostatics to solutions has given rise to many debates; an accurate and detailed analysis of this problem has been provided by Blaive and co-workers [S--8] _ For a fared charge distribution p&t placed in the dielectric cavity, classical electrostatics gives the work of assembling the charges of the system, at constant temperature and volume, through the relation
Ii’=&
f E-lfd7,
in terms of the vector fields E (electric field strength) and D (dielectric displacement) or, equivalentiy, through the relation:
where the total electrostatic potential Yis divided into its molecular and reaction components_ In general, instead of using (8) directly, the evaluation of the difference Wii with respect to a given reference model is introduced. We recall there have been several uses of the simplified expression
(9)
because the factor f corresponding to the first addend of the right-hand side of eq_ (8) is already incorporated in the hamiltonianWe could use the Same reference system as for eq. (9), obtaining
with Eo = C&S I$* f&Q
02)
for which all the terms are individually provided by the procedure shown in paper I_ The difference between eq. (11) and eq. (9) is given by the ~trodnction of the polarization of phi and by the inclusion of differences in the mean kinetic energy in the two systems. A defurition of Wf analogous to that given in eq. (IO) has been assumed also for the other states of the cycle_ The identification of Wi values with clearly defined thermodynamic quantities depends on the physical conditions in which the chargiug processes are performed. The work WI may be considered as done at stant temperature and density, and the state attained at the end of the process is in equilibrium. The work WI has thus the status of a free energy, and the difference Wol may be considered as the free-energy change due to the process of solution of gaseous MGS , apart from a numerical factor regarding the definition of appropriate reference states for gaseous MGS and for the solution. Similar considerations hold for ?U3 and for the differences WC3and %Vl 3, provided that the lifetime of ME)~ :b sufficiently Iong to ensure sofvent equilibration. States 2 and 4 of the cycle are not, by definition, in 79
Volume 99. number 1
equilibrium. It has been in fact assumed that suclt intermediate states arc followed by a relatively rapid reor~anisation of the mean distribution of the solvent around M. The defmition of the nature of W2 and It!+ pertains thus to the domain of non-equilibrium thermodynamics_ With a stricter definition of the esperimentaf conditions. also Wz and It; may be reduced to free energies. For many applications however it is not necessary to consider Iv2 and IU, (see below). The evaluation of internal energy changes. if req;lestcd. may be performed by implementing the computational scheme given in I with the numerical procedure tested in ref. f IO] for the classical case. WhiIe eq_ (I I) makes reference to the 0 state (gaseous hlGS plus dieIectric)_in fig. 1 state I has been employed as reference. In this way other terms, not accounted for by the electrostatic picture. cancel to a good atent. In particular we mention that the entropic term corresponding to the location of M in a fixed position of the solution. JS 11.~ been implicitly done in the definition of (I) (compare witlt tlte liberation free energy introduced by Ben Maim [ 111). cancels out. Cavitation energy contributions are reduced to a rcldtheiy sm,dl difference (equal to zero in the model cycle considered here). Wltat remains is mainly due to dispersion ~ontributioi~s and to changes in the vibrorotational contributions of hi to the free energy. Refercnce is made to a preceding paper [i2J for a discussion about the ewluation of these quantities. It will be sufficient to remark here that in examples considered in section 3 these contributions to the energy ch.mges of the system are f,rirly modest and they are neglected here.
3. Some numerical
esamples
With the ,rbovemcntioned simplifications we are non Jble to assess the energies considered in fig_ 1 for au &ctuaI molecular esnmple. In table 1 we report the results for the 1(n -+ rr* ) cycle of H2C0, calculated at the standard geometry 1131 with a STO-3G basis and m&ing use of the El-II’ method [ 141 for the evaluation of the excited-st,rte energy. The solvent is characterized by E = SO and ~2 = 1.777. The cavity, held fied, is cornposed of four spheres centered on the nuclei, with radii R, = 1.4. Rc = I.&Rn, =RrJz = 1.2 A. The first co1u1n1~of table I gives the change of Itt for the so
32 July I983
CHEMICAL PHYSICS LETTERS
Table 1 W~lucs for the different steps in the model cycle involving wrtical e.wzitation of IizCO to the ‘(n - n*) strife. (values in keai/niol) a)
I-2
2-3 3--J
1-l
104.72 -12.62 -8795 4-15
111.5s -1550 -S6.22 -9.83
6.83 -2.88 1.73 -5.68
98.79 -9392
a) It’-- = E-- -$A~P 2’ d-i. /I: absorption or emission encr+cs ~%ut~tlfd accord& to p&per I. Tile Wet V&IC (for it:dcfinitiorl see tfxe text) is -5.35 kcaljmol.
four steps considered in the cycle: a positive value corresponds to absorption of energy from the exterior and a negative one to release of energy to the exterior. Columns 2 and 3 give the decomposition of IQ into the two contponents set out in eq. (10). The results for Wij are reasonable: step 1 -+ 2 corresponds to the absorption of radiation; the evaluation of the absorption wavenumber performed according to the method of paper I gives iw = 34550 cm-l = 95.79 kcai/mol; step 3 --+4 corresponds to the emission of light; the emission wavenumber (see paper I) islzv=32850 cm-l = 9392 kcalfmol. The difference between I$$ and hv in the same process is due to differences in the physical models adopted here and in paper I. The cycle we have thus far considered may be further elaborated by including other electronic states, changes in the internal geometry of M, and changes in tfte cavity. Let us cousider firstly the question of the cavity at fised internal geometry of M, Supermolecule calculations on the system M-S, fir = I,& ___)indicate that when RI is an elicited electronic state there is in some cases a noticeable change in the M-S equilibrium distance. Some data on the l+CO-Hz0 system obtained with a basis set similar to that employed in the preceding calculations may be found in a paper by Iwata and hiorokuma [l 5) _Supplementing these reresults with other unpublished calculations of our group. it may be estimated that in the ‘(n 3 z*) state the mean distance between H,CO and the ftrst hydration shell in the carbonyi region is larger than in the GS by 0.3-0.4 A, The introduction of an equivalent enlargement in the cavity for the excited state influences states
Volume 99, number 1
CHEMICAL
PHYSKS
3 and 4 of our model cycle. The numerical effect of the enlargement of the&, radius by 0.3 A is the following: IV,_, changes to -9.83, W3_+ to -90-87 and W4_r to -4.03 kcaljmol. These values include also a small contribution due to the difference in the cavitation energy for the two cavity models considered here: according to Pierotti’s formulas [ 161 the enlargement of the cavity gives rise to an increase in the free energy of cavitation of 036 kcaljmol. Under the hypothesis that the solvent has a short relaxation time when compared with internal motion times of M and with the lifetime of Mu,, we may use W, and IV, determined at different internal geometries of M to define free-energy surfaces in the nuclear configuration space of M. The legitimacy of using freeenergy surfaces to study the dynamics of molecular systems has been clearly assessed [ 17]_ The conditions required for the definition of these surfaces do not represent a restriction with respect to the assumptions made above_ It is in fact sufficient to make formal reference to a statistical assembly of molecules M in the solute, i.e. to a sufficiently large specimen of the solution with the assumed characteristics,in order to define the height of the surface at any point of the contiguration space as a measure of the relative probability of that configuration (at a given temperature)_ When these surfaces have been determined, the intuitive considerations often employed in gaseous photochemistry [2] can be adopted. The application to these systems of detailed dynamical descriptions of the decay process, nowadays available for simple gas-phase systems, could also be investigated. For the case of the cycle involving the 1 (n--f n*) state of formaldehyde the free-energy surfaces may be sketched in terms of a parameter X connecting Iinearly the experimental [ 161 GS (A = 0) and EX (A = 1) geometries, thus reducing to one dimension the nuclear conformation space. The results are set out in fig_ 2. The most probable radiative transition 1(z*_* n) occurs in solution at h = 1.10. Le. at a geometry near but not equivalent to that of equilibrium for EX in the gas state (which in the approximation employed here occurs at A= 0.85). The emission wavenumber (method of paper I) is Bu = 22047.8 cm-r = 63.05 kcal/mol. The difference in energy WI@ = I .lO) - W3@ = 1 .lO) = -61.85 kcal/moI corresponds to that obtainable as W3r in table 1, but referred now to a different geometry of M. The difference W,(X= 0)- WI@= 1.10) =
LETTERS
23 July 1983
Fig,. 2. Schen~t~cd~~anl of the freetnergy chances for H&O along a coordinate in the nudear coniiguration space passing through the geometries of equilibrium for the GS and for the ‘(n-d ) state. The curves refer to states 1 and 3 considered in fis. I. GS geometry: C-O = 1.210 A, C-H = 1.103 A, LHCH = 1.121“; EX geometry: C-O = 1.323 A. C-H = 1.093 A, LHCH = 119”; bending angle 31”. The solvent uvity is modelled o,cr fourspheres: R0 ~lA(1_8forEX),R~=I_6.R~~, = RH2=l_2A_
-22.58 kcaI/mol corresponds to energy changes due to the relaation of the geometry of M in the GS and to the ensuing rearrangement of the solvent. %%en the excited state of M does not have a sufficiently long lifetime and/or the solvent is rather viscous, the dynamical behaviour of the systems cannot be reasonably appro_ximated in terms of a jump from the IV, to the IV, surface at the most favourable geometry_ These questions however lie beyond the limits of our simplified model.
References [l ]
N. hiatqa and T. Kubota, blolecuhr interactions and
electronic spectra (Dekker, New York. 1970). [ 21 J. hficltl, Top_ Current Chem. 46 (1974) l_ [ 31 R. Bonaccorsi. R. Cimiragha and J. Tomasi. to be published_ [4] S. Mirrtus. E. Saocco and J. Tom&, Chem. Phys_ 55 (1982) 117_ [S] B. Bkdw, Docteur es Science Thesis, Ah-Marseille III (1980).
Sf
Volume
99, number
1
CHEMICAL
PHYSICS
[6] B_Bl~ivrandJ_~let~cr,J_Phys.(~ris)42 (1981)1533. [7] B. Bfaiw and J. kfetzgcr, Physiu 119A (1983) 553. IS] B. Bltive and J. Metzger, Now. J. Chem., Submitted for publication. [9] J.G. Kirkwood, J. Chem. Phys. 2 (1934) 351. [lo] S. hliertus and J. Tomasi, Chem. Phys. 65 (1982) 239. ] II] A. Ben Naim, J. Phys. Chcm. 82 (1978) 792. [ 131 R. Bonxcorsi. C. Ghio and J. Tomasi, Current aspects of quantum chemistry 1981. cd. R. firbo (Else~icr. Amsterdam. 1982) p. 407. 1131 J.A. Poplc and 31. Gordon. J. Am. Chem. Sot. 89 (1967) 4253.
LETTERS
22 July
1983
[ 141 R_ hforokuma and S. Iwata, Chem. Phys. Letters 16 (1972) 192. [IS] S. Rata and K;. hforokuma, J. Am. Chem. Sot. 95 (1973) 7563. [ 161 G. Herzberg, hlolccular spectra and molecular structure, Vol. 3 (Van Nostrand, Princeton, 1967). [I71 K.J. Laidler and J-C. Pohnji. in: Progress in reaction kinetics, Vol. 3, ed. G. Porter (Pergamon. Otiord, 1965) p-l.
CHEMICAL
PHYSICS LEl-l-ERS
22 July 1983
ON THE FREE ENERGY CHANGES OF A SOLUTION IN LIGHT ABSORFTiON
OR EMISSION PROCESSES
R. BONACCORSI Istituto di Cldmim Quantisticned Energetica Molecolare de1 Ch’R. Via Risorginento 35.56100 Piss. Italy and
R. CIMIRAGLIA and J. TOMASI Zstituto di CJdmicaFisicn. Universirridi Pisa. 1% Risorgimento 35.56100 Pisa. Italy
Received 29 March 1983;in
final form 21 hiay 1983
A simple model to evaluate energy changes in dilute solutions produced by absorption or emission of light is proposed_ The computational scheme exploits some characteristics of the continuum model of liquids and is inserted into ab initio quantum-mechanical programs for isolated molecules. The model is addressed to evaluate changes in thermodynamic properties of the whale solution - while earlier models often consider the solute only -and it presentsa detailed decomposition of the phenomenon into successive steps.
1. Introduction
The processes of absorption and emission of light by a molecular solute in a solution cause several dynamical phenomena whose theoretical description requires a considerable effort. And the interpretation of solvent shifts on electronic spectra, though “prima facie” appropriate to energy transitions only, is confronted with these dynamical processes. The objective of getting manageable formulas of practical use has suggested drastic simplifications (for a detailed review see, e.g. ref. [l]). As far as the dynamics of the solute M is considered, the process is generally outlined by dividing it into separate steps, each treated in a time-independent formalism_ Attention is focused on vertical absorption from the ground state (GS) to an excited state (EX), at the GS equilibrium geometry, and on the emission from EX, in principle at a different geometry,selected on the basis of empirical considerations on the shape of the EX potential energy surface [2] _The dynamics of the rearrangement of the solvent molecules S is dealt with 0 009-26 14/83/0000-OOOO/% 03-00 0 1983 North-Holland
statistically (except for the cases in which it is necessary to introduce specifc interactions between solvent and solute, as occurs in the formation of excimers and in other intermolecular excitation transfers). Most methods make use of the continuum description of the solvent; in these models the dynamics of the solvent is represented by the changes in the reaction field due to the absorption (or emission) process, introducing a distinction between polarization effects due to the orientation of the solvent molecules (related to the macroscopic dielectric constant e) and polarization effects due to the induction of displacements in the electronic charge distribution of the solvent molecules (related to the square of the macroscopic refractive index n*)_ Recently we proposed a computational scheme [3] which translates the abovementioned features into the ab initio procedures of common use for gas-phase molecular calculations. In that paper (herafter called I) attention was focused on the evaluation of absorption and emission wave numbers at a given geometry of M, disregarding the energetic questions related to the 77
\rolumc 99, number 1
22 July 1983
CHEIIICAL PHYSICS LETTERS
other dynamical aspects, summarized above. We give here a complement to that scheme in which energy changes of the whole solution during the light absorption/emission processes are considered.
2. Theoretical aspects
01
=
qot(PGs4 -
(2)
3
In eq. (2) we have indicated the parameters on which (Tdepends, namely the GS charge distribution and the static dielectric constant E. We remark in addition that: oto&GS?
E) = o,,(PGS)
+ %&GS-
lz7) -
(3
The method referred to in paper I. which we shall not summarize here. was concerned with the application to the special case of the electronic band shift of a more general procedure [3] to deal with solvent effects_ It will be sufficient for the present purposes to recall that the solute-solvent interactions are introduced in the molecu@r hamiltonian by means of an additional operator I’, expressed in terms of a polarization surface charge distribution u accommodated on the cavity surface:
The total and inductive components of u can be computed separately, while the orientational component is defined as a difference(2) M absorbs a photon of the correct energy and the solute passes from M,, to ME,. The solvent esperiences the changes in the M charge distribution through its inductive part only, because the electronic transition time is muc11 shorter than the orientational relaxation time. The polarization vector is now Pz. related to
f3,t = @, + I;b _
02 = o,,@&
(1)
The charge distribution u is a function of the solute cl~irge distribution p5t (including both electron and nuclear contributions) and is due. according to the case. to the orientatiomd. inductive or total component of the polarization vector Pof the dielectric continuum simuking the solvent. The method of calculation of u depends on the spectroscopic process under consideration_ When the appropridte u. and then ?a_ is obtained. the difference in energy between the two states involved in the electronic transition is computed with current ab initio procedures. The ewluation of energy changes in the solution
produced by light absorption or emission requires some rearrangements of the computational scheme outlined above. JS we shall elucidate in the following schematic model example. Let us consider a dilute solution. for which the customary assumptions arc accepted (sufficient separation of the charge distributionsof Al and S, electronic excitation confined to hi. lack of dielectric saturation phenomena. etc.) and for which also the change in internal geometry of hl is negligible_ We consider now a simple cycle which passes through the following states: (1) The solute M in the GS is in equilibrium with the solvent_ The charge distribution pu gives rise to a polariration P, of the dielectric. accounted for by a surface charge distribution: 7s
+ o,,(,+Y-
n3-) -
(9
u, is introduced. via the corresponding operator, into the hamiltonian, and the absorption frequency is calculated using this hamiltonian. (3) hlEx has a sufficiently long lifetime to rearrange completely the solvent around itself. The solvent reaches a new polarization state P3.related to 03 = U,,&JEX. E) -
(5)
(4) Through an emission process (non-radiative decaysare neglected here) the CS is reached again. During this process only the inductive part of u is changed, and the calculation of the electronic level difference is performed by using a pa operator. corresponding to a polarization state P4 related to u4 = uor(PEX) + uind @GS 71z2) -
(6)
(5) Equilibrium of the solvent occurs and the same physical situation as in state 1 is reached. The evaluation of the electronic transition wavenumbers corresponding to steps 1 -+ 2 and 3 -+ 4 have been performed in paper 1 by considering the solvent as a bath, thus disregarding other energetic changes. The energy change in the whole cycle may be described with the diagram given in fig_ 1_ This diagram refers to a conventional space having as coordinates the energy and the state of polarization of the dielectric_ This last coordinate has not in the drawing a quantitative definition (even though when necessary a
Volume 99, number 1
CHEMICAL PHYSICS LETTERS
22 iuly 1983
which gives the difference of W with respect to a state composed of gaseous M (with the charge distribution &) and the separate dielectric containiig the appropriate cavity, assuming moreover that phf = & (i.e. disregarding polarization of M in the dielectric) [9,10] _ Returning now to our model, it is evident that a quantity analogous to eq. (8) and related to state 1 of our cycle is
ClO) F&_ 1. Schematic relationship of the four steps considered in the model cycle in s solvent pohrizationenergy coordinate frame. 1 refers to hfGs in equilibrium with the solvent; 2 to h&x with solvent not in equilibrium;3 to MEXin equilibrium with the solvent;4 to MGS with solvent not yet equilibrated.
Steps I--, 2 and 3 --+4 respectivelyinvolveabsorptionand emission of li$t.
more realistic scale could be elaborated). For a proper de~mit~ouof the energy it is convenient to consider the corresponding classical case_ Application of classical electrostatics to solutions has given rise to many debates; an accurate and detailed analysis of this problem has been provided by Blaive and co-workers [S--8] _ For a fared charge distribution p&t placed in the dielectric cavity, classical electrostatics gives the work of assembling the charges of the system, at constant temperature and volume, through the relation
Ii’=&
f E-lfd7,
in terms of the vector fields E (electric field strength) and D (dielectric displacement) or, equivalentiy, through the relation:
where the total electrostatic potential Yis divided into its molecular and reaction components_ In general, instead of using (8) directly, the evaluation of the difference Wii with respect to a given reference model is introduced. We recall there have been several uses of the simplified expression
(9)
because the factor f corresponding to the first addend of the right-hand side of eq_ (8) is already incorporated in the hamiltonianWe could use the Same reference system as for eq. (9), obtaining
with Eo = C&S I$* f&Q
02)
for which all the terms are individually provided by the procedure shown in paper I_ The difference between eq. (11) and eq. (9) is given by the ~trodnction of the polarization of phi and by the inclusion of differences in the mean kinetic energy in the two systems. A defurition of Wf analogous to that given in eq. (IO) has been assumed also for the other states of the cycle_ The identification of Wi values with clearly defined thermodynamic quantities depends on the physical conditions in which the chargiug processes are performed. The work WI may be considered as done at stant temperature and density, and the state attained at the end of the process is in equilibrium. The work WI has thus the status of a free energy, and the difference Wol may be considered as the free-energy change due to the process of solution of gaseous MGS , apart from a numerical factor regarding the definition of appropriate reference states for gaseous MGS and for the solution. Similar considerations hold for ?U3 and for the differences WC3and %Vl 3, provided that the lifetime of ME)~ :b sufficiently Iong to ensure sofvent equilibration. States 2 and 4 of the cycle are not, by definition, in 79
Volume 99. number 1
equilibrium. It has been in fact assumed that suclt intermediate states arc followed by a relatively rapid reor~anisation of the mean distribution of the solvent around M. The defmition of the nature of W2 and It!+ pertains thus to the domain of non-equilibrium thermodynamics_ With a stricter definition of the esperimentaf conditions. also Wz and It; may be reduced to free energies. For many applications however it is not necessary to consider Iv2 and IU, (see below). The evaluation of internal energy changes. if req;lestcd. may be performed by implementing the computational scheme given in I with the numerical procedure tested in ref. f IO] for the classical case. WhiIe eq_ (I I) makes reference to the 0 state (gaseous hlGS plus dieIectric)_in fig. 1 state I has been employed as reference. In this way other terms, not accounted for by the electrostatic picture. cancel to a good atent. In particular we mention that the entropic term corresponding to the location of M in a fixed position of the solution. JS 11.~ been implicitly done in the definition of (I) (compare witlt tlte liberation free energy introduced by Ben Maim [ 111). cancels out. Cavitation energy contributions are reduced to a rcldtheiy sm,dl difference (equal to zero in the model cycle considered here). Wltat remains is mainly due to dispersion ~ontributioi~s and to changes in the vibrorotational contributions of hi to the free energy. Refercnce is made to a preceding paper [i2J for a discussion about the ewluation of these quantities. It will be sufficient to remark here that in examples considered in section 3 these contributions to the energy ch.mges of the system are f,rirly modest and they are neglected here.
3. Some numerical
esamples
With the ,rbovemcntioned simplifications we are non Jble to assess the energies considered in fig_ 1 for au &ctuaI molecular esnmple. In table 1 we report the results for the 1(n -+ rr* ) cycle of H2C0, calculated at the standard geometry 1131 with a STO-3G basis and m&ing use of the El-II’ method [ 141 for the evaluation of the excited-st,rte energy. The solvent is characterized by E = SO and ~2 = 1.777. The cavity, held fied, is cornposed of four spheres centered on the nuclei, with radii R, = 1.4. Rc = I.&Rn, =RrJz = 1.2 A. The first co1u1n1~of table I gives the change of Itt for the so
32 July I983
CHEMICAL PHYSICS LETTERS
Table 1 W~lucs for the different steps in the model cycle involving wrtical e.wzitation of IizCO to the ‘(n - n*) strife. (values in keai/niol) a)
I-2
2-3 3--J
1-l
104.72 -12.62 -8795 4-15
111.5s -1550 -S6.22 -9.83
6.83 -2.88 1.73 -5.68
98.79 -9392
a) It’-- = E-- -$A~P 2’ d-i. /I: absorption or emission encr+cs ~%ut~tlfd accord& to p&per I. Tile Wet V&IC (for it:dcfinitiorl see tfxe text) is -5.35 kcaljmol.
four steps considered in the cycle: a positive value corresponds to absorption of energy from the exterior and a negative one to release of energy to the exterior. Columns 2 and 3 give the decomposition of IQ into the two contponents set out in eq. (10). The results for Wij are reasonable: step 1 -+ 2 corresponds to the absorption of radiation; the evaluation of the absorption wavenumber performed according to the method of paper I gives iw = 34550 cm-l = 95.79 kcai/mol; step 3 --+4 corresponds to the emission of light; the emission wavenumber (see paper I) islzv=32850 cm-l = 9392 kcalfmol. The difference between I$$ and hv in the same process is due to differences in the physical models adopted here and in paper I. The cycle we have thus far considered may be further elaborated by including other electronic states, changes in the internal geometry of M, and changes in tfte cavity. Let us cousider firstly the question of the cavity at fised internal geometry of M, Supermolecule calculations on the system M-S, fir = I,& ___)indicate that when RI is an elicited electronic state there is in some cases a noticeable change in the M-S equilibrium distance. Some data on the l+CO-Hz0 system obtained with a basis set similar to that employed in the preceding calculations may be found in a paper by Iwata and hiorokuma [l 5) _Supplementing these reresults with other unpublished calculations of our group. it may be estimated that in the ‘(n 3 z*) state the mean distance between H,CO and the ftrst hydration shell in the carbonyi region is larger than in the GS by 0.3-0.4 A, The introduction of an equivalent enlargement in the cavity for the excited state influences states
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3 and 4 of our model cycle. The numerical effect of the enlargement of the&, radius by 0.3 A is the following: IV,_, changes to -9.83, W3_+ to -90-87 and W4_r to -4.03 kcaljmol. These values include also a small contribution due to the difference in the cavitation energy for the two cavity models considered here: according to Pierotti’s formulas [ 161 the enlargement of the cavity gives rise to an increase in the free energy of cavitation of 036 kcaljmol. Under the hypothesis that the solvent has a short relaxation time when compared with internal motion times of M and with the lifetime of Mu,, we may use W, and IV, determined at different internal geometries of M to define free-energy surfaces in the nuclear configuration space of M. The legitimacy of using freeenergy surfaces to study the dynamics of molecular systems has been clearly assessed [ 17]_ The conditions required for the definition of these surfaces do not represent a restriction with respect to the assumptions made above_ It is in fact sufficient to make formal reference to a statistical assembly of molecules M in the solute, i.e. to a sufficiently large specimen of the solution with the assumed characteristics,in order to define the height of the surface at any point of the contiguration space as a measure of the relative probability of that configuration (at a given temperature)_ When these surfaces have been determined, the intuitive considerations often employed in gaseous photochemistry [2] can be adopted. The application to these systems of detailed dynamical descriptions of the decay process, nowadays available for simple gas-phase systems, could also be investigated. For the case of the cycle involving the 1 (n--f n*) state of formaldehyde the free-energy surfaces may be sketched in terms of a parameter X connecting Iinearly the experimental [ 161 GS (A = 0) and EX (A = 1) geometries, thus reducing to one dimension the nuclear conformation space. The results are set out in fig_ 2. The most probable radiative transition 1(z*_* n) occurs in solution at h = 1.10. Le. at a geometry near but not equivalent to that of equilibrium for EX in the gas state (which in the approximation employed here occurs at A= 0.85). The emission wavenumber (method of paper I) is Bu = 22047.8 cm-r = 63.05 kcal/mol. The difference in energy WI@ = I .lO) - W3@ = 1 .lO) = -61.85 kcal/moI corresponds to that obtainable as W3r in table 1, but referred now to a different geometry of M. The difference W,(X= 0)- WI@= 1.10) =
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23 July 1983
Fig,. 2. Schen~t~cd~~anl of the freetnergy chances for H&O along a coordinate in the nudear coniiguration space passing through the geometries of equilibrium for the GS and for the ‘(n-d ) state. The curves refer to states 1 and 3 considered in fis. I. GS geometry: C-O = 1.210 A, C-H = 1.103 A, LHCH = 1.121“; EX geometry: C-O = 1.323 A. C-H = 1.093 A, LHCH = 119”; bending angle 31”. The solvent uvity is modelled o,cr fourspheres: R0 ~lA(1_8forEX),R~=I_6.R~~, = RH2=l_2A_
-22.58 kcaI/mol corresponds to energy changes due to the relaation of the geometry of M in the GS and to the ensuing rearrangement of the solvent. %%en the excited state of M does not have a sufficiently long lifetime and/or the solvent is rather viscous, the dynamical behaviour of the systems cannot be reasonably appro_ximated in terms of a jump from the IV, to the IV, surface at the most favourable geometry_ These questions however lie beyond the limits of our simplified model.
References [l ]
N. hiatqa and T. Kubota, blolecuhr interactions and
electronic spectra (Dekker, New York. 1970). [ 21 J. hficltl, Top_ Current Chem. 46 (1974) l_ [ 31 R. Bonaccorsi. R. Cimiragha and J. Tomasi. to be published_ [4] S. Mirrtus. E. Saocco and J. Tom&, Chem. Phys_ 55 (1982) 117_ [S] B. Bkdw, Docteur es Science Thesis, Ah-Marseille III (1980).
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[6] B_Bl~ivrandJ_~let~cr,J_Phys.(~ris)42 (1981)1533. [7] B. Bfaiw and J. kfetzgcr, Physiu 119A (1983) 553. IS] B. Bltive and J. Metzger, Now. J. Chem., Submitted for publication. [9] J.G. Kirkwood, J. Chem. Phys. 2 (1934) 351. [lo] S. hliertus and J. Tomasi, Chem. Phys. 65 (1982) 239. ] II] A. Ben Naim, J. Phys. Chcm. 82 (1978) 792. [ 131 R. Bonxcorsi. C. Ghio and J. Tomasi, Current aspects of quantum chemistry 1981. cd. R. firbo (Else~icr. Amsterdam. 1982) p. 407. 1131 J.A. Poplc and 31. Gordon. J. Am. Chem. Sot. 89 (1967) 4253.
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1983
[ 141 R_ hforokuma and S. Iwata, Chem. Phys. Letters 16 (1972) 192. [IS] S. Rata and K;. hforokuma, J. Am. Chem. Sot. 95 (1973) 7563. [ 161 G. Herzberg, hlolccular spectra and molecular structure, Vol. 3 (Van Nostrand, Princeton, 1967). [I71 K.J. Laidler and J-C. Pohnji. in: Progress in reaction kinetics, Vol. 3, ed. G. Porter (Pergamon. Otiord, 1965) p-l.