VIBRATIONALENERGY TRANBFER IN Hz
+ He
400 1
Vibrational Energy Transfer in the Hydrogen Molecule-Helium Atom Systemla
by Hyung Kyu Shin Department of Chemistry,'b University of Nevada, Reno, Nevada
89607
(Received M a y 24, 2971)
Publication costs assisted by the U.S . Air Force O f i c e of Scientific Research
+
Vibrational energy transfer in HI He is studied by use of t'he accumte a priori interaction potent'ial. An expression for the vibrational transition probabilities is obtained by solving the Schrodinger equation for the perturbed oscillator states. The expression explicitly shows the dependence of the trailsition probability on molecular orientations, oscillator anharmonicity, and multiquantum transitions. Detailed numerical study for the 0 + 1 vibrationaltransition in the electron-volt range is carried out to show the dependence of the transition probability on the relative collisioii energy and the orientation angle. At higher energies, the introduction of the aiiharmonicity correction decreases the vibrational t,ranfiit,ionprobabilit'y by la,rge factors. The temperature dependence of the bransition is also discussed with the angle and energy-averaged t'ransitian probability, Vibrat'ion-rotation energy transfer is not considered, but the amount of energy transfer to the anharmonic oscillator is calculated as a functionof the initial orient'ationangle.
Introduction I n recent years, with the advent of large memory highspeed computers, accurate a priori interaction potential energy functions have become available for the calculation of probabilities of molecular energy transfer.2-6 Among them are the relatively simple forms for Hz He computed by Krauss and X e s z band by Gordon and Secresta5 Recently, Lester4obtained a lengthy function Hz by employing accurate self-consistent for L i t field wave functions. These functions, which show detailed orientation dependences, are obtained for the regions which are appropriate for the study of vibrational excitations. Since such exact potential functions are becoming available, we can now make rigorous calculations of vibrational transition probabilities and related quantities. Rfics6 used the H2 $- He potential in his numerical evaluation of the 0 + 1 vibrational transition probability within the framework of the distorted wave approximation (DIVA)) and compared the result with that of model potentials to show the inadequacy of the He collision. Allatter forms in describing the €I2 though the calculation revealed new features of the transition process, it is generally known that the DWA cannot be used to calculate the vibrational transition probabilities in the region of high collision where the approximation may lead to the situation that the probabilities exceed unity. The DWA, which i s simple to use, normally gives acceptable results for low probabilities. The a priori Hz He potential function is in a simple form, which can be readily used for accurate calculation of the transition probabilities. I n the present paper, we developed a method for the calculation of vibrational transition probabilities for H2 He. The method involves the solution of the Schrodinger equation describing the perturbed oscillator states in an explicit form. We shall use the potential function obtained by Krauss
+
+
+
+
+
and Mies. Although numerical calculation of the transition probabilities will be shown, we shall consider the development of the method in detail; the development will be made such that the method can be readily used with other collision systems for which accurate forms of the interaction potentials are available.
Potential Energy Functions The a priori interaction energy for Hz mined by Krauss and hlies is Il'(x,q,e)
=
C exp(-
all:
+ He deter-
+
+~
q[A@) ) B(8)qI (1)
where q is the displacenient of the oscillator from its equilibrium position Re,x the distance between the center of mass of Hz and He, a = 1.86176 au-l, cy1 = 0.3206 au-2, C = 198.378 eV, A(8) = 1.0 f 0.30124 0.21617 cos2 8 au-l. cos2 8, and B(8) = -0.59932 This function accurately represents the Hz He interaction for the ranges0 I q R, 5 2 au and 2.5 5 x I 3.8 au. I n using eq 1for the formulation of vibrational transition probabilities, we face mathematical difficulty due to the appearance of the q dependence both in the preexponential and exponential parts. However, this difficulty can be avoided by expanding exp(wq) in a
+
+
+
(1) (a) This work was supported by the Directorate of Chemical Sciences, the U. S. Air Force Office of Scientific Research, under Grant AFOSR-68-1354. (b) Theoretical Chemistry Group Contribution No, 9-1033. (2) (a) C. S. Roberts, Phys, R e v , , 131, 203 (1963); (b) M.Krauss and F aH.X e s , J . Chem. Phys., 42,2703 (1965). (3) W.A . Lester, ibid., 53, 1511 (1970); 53, 1611 (1970). (4) TV. A . Lester, ibid., 54, 3171 (1971). (5) M. D. Gordon and D. Secrest, ibid., 52, 120 (1970). (6) F. H. Vies, ibid., 42, 2709 (1965). (7) K. Takayanagi, A d % a n .M o l . At. P h y s . , 1 , 149 (1965). (8) D. Rapp and T. Kassal, Chem. Rev., 69, 61 (1969). (9) D. Serrest and B. R . Johnson, J . Chem. P h y s . , 45,4556 (1966).
T h e Journal of Physical Chemistry, V o l . 76, No. 26, 1971
IIYUNG KYUSHIN
4002
power series and taking the first several terms. For the range in which eq 1 is valid the exponential part exp' (aleq) lies in the range from exp(OS015q) to exp. (1.2182q). If we take p as large as +0.6 au, these two limits are 1.6175 and 2.0771, respectively. In the expansion, when we take the first four terms of the upper limit, exp(0,7309) 'v 2.0631 while its exact value is 2.0771. On the other hand, for q = -0.6 au, exp. (-0.7309) 'u 0,4711 while its exact value is 0.4514. The equilibrium value Re is 1.4 au, and probably the range of the displacement of the vibrational amplitude of the displacement of the vibrational amplitudes of interest is +0.3 au. Therefore, the error caused by the expansion of exp(a1xq) should be very small, and to a good approximation we can express the exponential part as exp(alw) = 1
+ alzq + ' / z ( a ~ x ) t~ q ~ ' / / ~ ( a l x )+ ~q~
'/2,(~21x)~q~ (2)
With this expression we now write the modified interaction potential as
C(q) = 1/zi14~2qz - '/2Mw2dq3
+qq1q + [l/zA(e)(alx)2 + + (aIz) + l/zB(q(WPIY~ +
+ 7/24LTl~242q4(6)
The second and third terms on the right-hand side are assumcd to reproduce the anharmonicity of the molecule. Therefore we shall rcplace U ( q ) in the Hamiltonian by ' / 2 Mu2q2 21+34 X,qj, where ha = - il/lw24 and X4 = 7/24 n/Iu2d2. We then have
+
-9
H
U(s,q,O) = CA(B) exp(-m) i-
c exp(--4{
potential cncrgy of thr oscillator, Such a potctitial must rcproducc thc oscillator's anharmonicity, which might exert an importarit influcnce on the vibrational energy transfer. Potentials such as the llorsc function U ( q ) = D,[1 - ~ x p ( - d q ) ]can ~ hc used hcrc, whcrc 4 is a rangc parameter to be deterrnincd and D,is thc dissociation encrgy. However, thc Slorsc function is difficult to handle in the method dcscribcd bclow, but without causing any serious error wc can expand it about y and take thc first scvcral trrms. By defining the force constant by [dZP(p)/dq2],=~,wc can then approximate the Hz molecule by an anharmonic oscillntor with the potential
=
rJ- + ' / z M W 2 q 2 2M
- F(t)q
+
4
x,q* j = 3
+
4
c
[A(@)alZ
2=2
74t)Y'
(7)
a l ~ ( e ) ~ ~ ~ 2
[1/4e)
where --F(t)
= ql(t).
3
[1/24~(e) (011%)~ + 1/613(e) (a1~)3~q4 }=
+ '"(',qJe)
u(z>o,e)
Perturbed Wave Function (3)
where the q-dependent part C7'(x,q,B)is responsible for vibrational transitions, while U(x,O,e) essentially controls the relative translational motion of the collision partners. We shall express the q-dependent terms in the form 4
uYg,q,e) =
C 1v , ( x ) ~
(4)
We are therefore concerned with the interaction of Hz with He through the perturbation energy 2qt (x)p', which explicitly includes vibration-rotation coupling. I n setting up the problem, we shall parametrize x in the time t, so that the sum can be considered to be a time-dependent perturbation energy. The Hamiltonian takes the form (5)
where p is the momentum and M is the reduced mass of the oscillator and U(y) is the intramolecular potential function. If we assume the harmonic motion of the oscillator, then the latter function is simply 1/2Mu2q2, where u is the vibrational frequency. In studying vibrational energy transfer problems, we therefore need to know, in addition to U(x,q,e), the intramolecular T h e Journal of Physical Chemistry, Vol, 76,No. $6, 1971
The oscillator is perturbed by the energy U'(z,p,e) during the course of collision, and w e now need to find thc wave function representing the perturbed state. Since the quantum system under the influence of U' [x (t),q,e] evolves in an exactly predictable manner, we can determine the wave function $ ( t ) representing its dynamical state at time t by specifying $(to) for the initial state a t t o , The wave function $ ( t ) then represents the perturbed state and can be obtained by solving the Schrodinger equation ifi$(t)
H$(t)
(8)
where the Hamiltonian is given by eq 7. If we discard the two sums in eq 7, the subsequent solution of eq 8, which is well known, will describe the case of the forced harmonic oscillator, We should therefore be able to take into account explicitly the effect of oscillator anharmonicity on the solution $(t) with the j sum. The appearaiice of the sums in eq 7 greatly complicates the solution of eq S. T o facilitate the solution, we introduce the operators a and a+, which are hermitian conjugates of each other satisfying the commutation relation [a,a+] = 1. The position variable p and the mo(10) I. I. Gol'dman and V. D. Drivchenkov, "Problems in Quantum Mechanics," Addison-Wesley, Reading, Mass., 1961, pp 103-106. Also see D. ter Haar, "Selected Problems in Quantum Mechanics," Academic Press, New York, N. Y., 1964, pp 152, 153. (11) C.E. Treanor, J. Chem. Phys., 43,532 (1965); 44,2220 (1966).
VIBRATIONAL ENERGY TRANSFER IN Hz
+ He
4003
mentum p are linear combinations of these operators1°,’2
By equating the coefficients of the operators N, a+, and a of the both sides of this equation, we find
(9)
In terms of these operators we can write the Hamiltonian as
where N = a+a. By use of the commutation relation and the identity [(a+)’, a ] = -%(a+)”-’,we can expand (a a+)iand (a a+)! in the form Z m , n cm,n(a+)man; e.g., (a a+)2= a2 2a+a a + 2 1. Such expansions will be used in obtaining eq 14-17 from eq 13 below. We shall first look for the solution of eq 8 in the The coefficients of the operators which are in higher orform1 O , 1 l t18 ders of a+, a, and N appeared on the right-hand side of eq 13 are set equal to zero, but the resulting relations do not contribute to the formulation of the relevant equations given above. I n deriving the differential We assume the initial wave function +(to) by the harequations! eq 14 has been used to simplify the three monic oscillator wave function and write it in the other equations. It is obvious that the solution of eq form Gm(q), m representing the initial oscillator state. 15 is the complex conjugate of that of eq 16. With the Therefore, eq 12 can be obtained as a linear combination initial conditions ( t o = - a), h ( - m ) = 0, f(- a) = of the unperturbed harmonic oscillator wave function 0, and c(- a ) = 1, we find the solutions with the coefficients determined by the time-dependent energy terms as well as the anharmonicity terms. The square of the coefficient of a particular state (say n) is then the probability that the oscillator is in the state n at time t; the probability at t = *, p,,, can be obtained by evaluating the coefficient a t t = m , With the time derivative of eq 12 and by use of the commutation relation and [(a+)’, a ] = -n(a+)‘-l in the expanded formloof exp [f(t)a+lexp [g(t)a]exp [ h ( t ) N ] , we find the following equation14from eq 8
+
+
+
+
+
+
+
iK(h(t)N
+
+ CfO) - h(t)S(t)la+ + [ k ( t ) +
(12) A. Messiah, “Quantum Mechanics,” Vol. I, North-Holland Publishing Co., Amsterdam, 1968, Chapter 12. (13) P. Pechukas and J. C. Light [ J . Chem. Phys., 44, 3897 (1966) 1 solved the problem of the linearly forced harmonic oscillator in terms of the operators a + and a. (14) H. Shin, Chem. Phys. Lett., 5 , 137 (1970).
The Journal
of
Physical Chemistry, Vol. 76, N o . 86, 1971
4004
HYUNGKYUSHIN
K(w/s,r/m,n). However, there can be more than one term contributing to the final state n so that the squaring should include the sum of all such terms. For example, for the transition 3 * 4, there are four terms leading to the $d state, namely ( s , ~ ) = ( O , l ) , (lJ2), (2,3), and (3,4), We generalize this situation to obtain the final expression of the vibrational transition probability as P,,
With the solutions for the coefficients given above, we +(t) as a particular combination of the unperturbed harmonic oscillator wave functions. The coefficients of such a combination will depend on the functions given by eq 18-21. For the wave function t,bm(q) we have the following recursion relations12
now solve eq, 12 for
a++m = (m a$,
=
+ 1)%,
+1
ml”t,bm- (m# 0 ) ; a+o = 0
(22)
=
(c(a)121f(m)12’n
-“‘m!n! x
in which the s sum is from zero to niin(m,n), the lower of m or n. We must note that this equation satisfies the principle of detailed balance P,, = P,, and that the probability is conserved; o&m, P,, = 1. For the oscillator initially in the ground state, eq 26 reduces to the simple form
(27) which can also be obtained from eq 25a. From eq 19 and 21, we have
(23)
Ngm =
(24) Since the operator a + transforms the state m into the state m 1, a + may be called an “excitation” operator, The operator a, on the contrary, transforms the state 1, thus acting as a “deexcitainto the lower state m tion” operator. Hence, in ey 12 each operator in the exponent will operate on #,(q) to generate the wave functions $O(S), # ~ ( q ) #, e ( d , . ., #n(q), . . -. BY expanding the exponential part of eq 121° and by generating the functions with the recursion relations given above, we obtain the wave function p(1) in the form
+
-
I
[s(t)18 -
+(t) = c(t) exp[mh(t)] s = o
;+
s) !
where AE may be defined as the amount of the energy transferred to the oscillator driven by the perturbation energy Zv%(t)q$, We note the appearance of the anharmonicity effect Xs in AE while X4 does not. If we set q 2 ( t ) , q 3 ( t ) , and the anharnionicity term Xa to zero, then the expression of AE reduces to the well known form16
s!
pm - s f r
where in the second relation n represents ( m - s For the oscillator initially in the lowest state m this equation reduces to
+ r). =
0,
Transition Probability The probability of the m 3 n vibrational transition can be obtained from eq 25 by squaring the coefficient The Journal of Physical Chemistry, Vol. 76, N o . 16,1971
for the harmonic oscillator driven by the force F ( t ) . For an explicit calculation of P,, we need to find the time dependence of y(x)’s from the x-t relation by solving the equation of motion as
(15) D. Ra,pp, J . Chem. Phys., 32,735 (1960).
~ T I I i I I A T I O K A LENERGY TRANSFER I N Hz
+ He
4005
where p is the reduced mass of the collision system, E the relative translational energy, and x* the largest root of E - U(s,cy,~) = 0. This equation determines the trajectory of the relative motion as a function of .9 and q, but we neglect the cffect of the oscillator’s displacement on the trajectory. Since we wrote
v(x,g,e)
=
C 4 e ) exp(--(yX)
+ U’(x,g,e)
following calculation section, we will find that this expansion is satisfactory. Then, thc amount of energy transfer can be given in the form AE =
‘sa
2M!
3fi
dt exp(iwt)
-m
(3)
the potential function appropriate for determining the trajectory can be taken to be CA(B) exp(-ax), and use it in eq 31. The essential part of the integrals in eq 27 and 28 will then have the form
exp [ - ax(t)
Jm -a
+ iwtldt
which can be computed by contour integration.16 As 0, the potential energy U(x,O,8) tends to CA(8), where C = 198.378 eV, Since A(8) is not significantly different from unity, this limiting value lies in a strongly repulsive region. By defining the collision time as1’
2 +
where the explicit forms of F(1), ?S(t), and q ~ ( t )are, respectively
F ( t ) = -c[a,A(e)x
+ B(e)]exp(-ax)
+ d ( 8 ) 2 1 exp(--)
d t ) = C[l/zA@)dx2 qa(t) =
we can solve eq 31 as
t
= ir
-
i -(-) a u 2p
(33)
C[i/&3)a1353
+ 1/2B(B)~12x2] exp(-aaz)
The right-hand side of these relations will be converted into the corresponding time-dependent forms by use of the relation 2 =
or
--1n{[3-]’/2 2 a
CA(8)
1
a(r
}
+ it)
(37)
and eq 34 itself. The resulting equation will contain integrals of the forms I n eq 32 the lower limit 50 represents the distance at which U(z,O,B) becomes strongly repulsive. A different choice of this integration limit would not alter the result; the necessary condition for the choice is U(z,,O,O) >> E. For U(x:,,e)= CA(0) exp( - ax)?the integration in eq 32 is trivial; the result is simply (35) Since the collision time is determined from the relative motion of the incoming atom with respect to the center of mass of the oscillator, the situation that the collision time is independent of the angle 8 is certainly expected in the present approximation. In eq 28 and 29 the term
appeared in the exponent of AE is due to the term proportional to q2 of U(x,q,B). The appearance of this term is seen in eq 14 indicating that the frequency varies with the time,l* i.e., w q2(t)/Mw. Except a t very high collision energies, the time-varying term is small, so that we can expand this exponential part in a power series and take the first three terms. I n the
+
and
where /3 and y are integers. Since there is a branch cut from i~ along the imaginary axis to i a ,we cannot simply replace the above integrals by their residues at the pole. Therefore, we replace the integration by an integration along a contour, which extends from + i a, encircles the singular point, i ~ ,and then follows a branch cut in the complex t plane.16 The latter form of the two integrals given above eventually reduces to the former when a successive use of the L’Hospital rule is made before letting t -+ ir. Carrying out the integration, we find
(16) E. E. Nikitin, Opt. Spektrosk., 6, 141 (1959); English transl., Opt. Spectrosc., 6,93 (1959). (17) H. Shin, J . P h y s . Chem., 73,4321 (1969). (18) A. Zeleohow, D. Rapp, and T. E. Sharp, J . Chern. Phys., 49,
286 (1968).
T h e Journal of Physical Chemistry, “01. 75, No. 96,2971
4006
HYUNGKYUSHIN t
The inkgral
s_..v a ( W
J--vz(t)dl may be nritten in the form t
t
=
J-m [‘IzCA(@)al2xZ + aiCB(0)~lCXP( - aX)dt (39)
which can be explicitly evaluated n hen dt is rcplaccd by dx through eq 37. The rcsult can bc readily converted back to a timc-dependent form by use of cq 34 for the evaluation of AE. We note that the leading term of the anharmonicity in eq 36 appears as X3
exp(iut)dt
which is an integral representation of the 6 function and vanishes in this case. Therefore, the first nonvanishing tcrm of the anharmonicity contribution is
l;m lt
A3
exp(iwt)dt
vz(t’)dt’
By use of eq 35 followed by lengthy but elementary operations and simplifications, we finally obtain the following expression for the amount of energy transfer
of f h c ovcvall interaction docs riot nppcwr in thc abovt. cxprcssion of AI$, L! hilt iEic anglc-dcpctidctit futicl ions A(e) arid B ( e ) m t w in u complicatcd way. Wr also note that in ihc function (/(e), thc first term is due to B(6)q of the o v ~ r d lpotential while thc second term results from A(0)alxq which is from the first q-dcpcndent Icrm of the expansion of cxp(alxq). Thrsc trio terms arc the most important part of thc potcritixl causing vibrational transitions, ie., g(0) ‘v [l -b 2(da)A(@/B(@ 12* Effects of Anharmonicity and Multi-quantum Transitions I n the above derivation we have discussed the cffcct of the oscillator anharmonicity on vibrational cncrgy transfer. Honcvcr, the cffcct was not fully accountcd for in the derivation of the vibrational transition probability. Equation 36 shows that if v2(t) were zero, tho anharmonicity correction would vanish, but the anharmonicity should contribute to the vibrational energy transfer even if vz(t) is zero. The form of $(t) given by eq 12 is not sufficient for a rigorous calculation of the anharmonicity since i t does not contain all the operators which act on the initial wave function to generate the anharmonicity. The leading part of the anharmonicity correction is proportional to X8(a a+)3, which can be expanded as X3(a3 a f 3 3aat2 3a2at - 3a+ - 3a). I n the expansion aa+z and a2af also act as one-quantum transition operators, but only thc operators a and a + were included in the above derivation of Pmm.The cubic operators a3 and a+3are thrccquantum deexcitation and excitation operators, respectively. For the oscillator initially in the ground state, a2a+and a3 should be discarded since they generate the $-I and states. I n the quadratic tcrm, (a a + ) z can be expanded as a2 a+2 2a+a 1, where a2and a+* are two-quantum transition operators, but these operators were not included in the above derivation. Therefore, a more rigorous form of the wave function is
+ +
+ +
where
Mw
+
+
The lengthy expression for g(0) is due to the inclusion of higher-order terms in Q in the Hamiltonian. For practical purposes, higher-order terms in ( a l / a ) , particularly those containing p / M u 2 and/or Ti/Mu, can be neglected. Our problem of finding an explicit solution of P,, is now completed with eq 26, 25, 30, and 40, and numerical computation AB, and in turn P,, as a function of E and e is simple. The energy constant C The Jourrkal of Phgrsieal Chemistrgr, VoZ. 7 6 , No. $6, 1971
+
+
+
This expression includes the one-quantum transition operator a a i 2 responsible for the leading effect of the anharmonicity and the two-quantum transition operator a+2. The latter operator represents the leading effect of multiquantum transitions. For the oscillator initially in the ground state, eq 41 contains all important information needed to determine the vibrational transition probability. By expanding the exponential operators and by generating the functions with the recursion relations as above, xve obtain a lengthy cxpression for the wave function
VIBRATIONAL ENERGY TRANSFER IN Hz
[v(t)l2 (m
m
+@) =
z=o
I!dm!(m + I)
m+L+ZJ: s=o
[f(t) 1' r!
(m
2 - 4
r=O
+ 21)!
+ He
+ +
(m
The fifth and sixth terms, which contain the effect of multiquantum transitions and oscillator anharmonicity, can be converted into
X
[g(t)]S (m I 2k)! -3-4?%+1+2k-4!
+ 1 + 2k - + ?")!
4007
x
and
8
+ I + 2k - s)!
+m+
1+21--s+r
C C C C K(tIl,k,s,+w)+, Z
L
S
(42)
V
In the second relation n, the final state, represents (m 1 2k - s T ) , and the sums are arranged in the order of their generation from the initial state #m. Because of the relation n = m 1 2k - s T , one of the indices in the middle group of the coefficient K
+ +
We thus add these two terms (after multiplication by ik) to the left-hand side of eq 13, and obtain
+
+ +
+
need not be determined in carrying out the summations. If we also introduced the operators az and ala+ in eq 41, there will be two additional sums appearing in eq 42. Such a treatment should, of course, be made in a more rigorous calculation. However, in the present paper, the inclusion of a + 2and aa+2in +(t) is sufficient in determining the leading effects of multiquantum transitions and anharmonicity. By evaluating the time dependent functions in eq 41 at t = 0 0 , we can write the transition probability as
and
from the coefficients of and a+2, respectively. For the 0 -+ 1 vibrational transition, the transition probability is then
pol
= lirn/c(t)]Z/f(t)/2jl t--L m
+ 2 f ( t ) v ( t )exp[h(t)
112
(47)
where higher-order terms (((1) in the square brackets have been neglected; such terms would become important in high energy collisions, however. Since the function u( m ) can be obtained from eq 45 as
rn+1+2k
c
[g(m)]yf(a,)]n--------k+4(rn
s!(m
+z+
2k)!
+ I + 21c - s)!(n- m - 1 - 2k + s)!
the transition probability becomes
(43) which reduces to eq 26 when v ( m ) and u ( m ) are set equal to zero. To determine the functions v ( t ) and u(t), we need to obtain the time derivative +(t) as follows
(49)
where the second term in the square brackets represents the effect of the anharmonicity. The expression can also be expressed as
exp(-
(44)
g)
(50)
For X3 = 0, this expression reduces to the well-known formsPol(E) = (AE/fia)exp( - A E / f i w ) . The energy transfer AE derived above depends only on the initial relative translational energy and not on the final relative energy. This dependence results because the trajectory x ( t ) calculated above is independent of the collision system's final state. ThereThe Journal of Physical Chemistry, Vol. 76, N o . 26, 1971
HYUNGKYUSHIN
4008 Table I : Calculated Values of POI( E )
1 2 3 4 5 6 7 8 9 10 12 15
18 20 a
(lo)@ 8 . 4 6 (10) (5) 2.82 ( 5 ) (4) 4.75 (4) 2.13 (3) (3) 5.82 (3) 5 . 6 1 (3) 1.12 (2) 1.16 (2) 1.80 ( 2 ) 1.83 (2) 2.59 12) 2.68 (2) 3.60 (2) 3.50 (2) 4.54 (2) 4.42 (2) 6 . 2 5 (2) 6.15 (2) 7.86 (2) 7.94 (2) 7.94 (2) 8 . 0 3 (2) 6.94 (2) 7.19 (2) 8.77 2.92 4.92 2.20
7.47 (10) 2.49 ( 5 ) 4 , 2 0 (4) 1.88 (3) 4.96 (3) 9.91 (3) 1.60 (2) 2.32 (2) 3.14 (2) 4.01 (2) 6.66 12) 7.52 (2) 8 . 1 7 (2) 7.82 (2)
5.72 (10) 1.91 (
