Ab Initio Molecular Orbital Calculations Of Reduced Partition Function .
Ab initio Molecular Orbital Calculations of Reduced PartitionFunction Ratios of Polyboric Acids and Polyborate AnionsTakao OiDepartment of Chemistry, Sophia University, 7-1 Kioicho, Chiyoda, Tokyo 102-8554, JapanReprint requests to Prof. T. O.; Fax: 81-3-3238-3361Z. Naturforsch. 55a, 6 2 3 - 6 2 8 (2000); received March 18, 2000Molecular orbital calculations at the HF/6-3 lG(d) level were carried out for polyboric acids and polyborate anions up to a pentamer to estimate their " B -to- B isotopic reduced partition function ratios( R P F R s ) and examine the additivity of logarithms of RPFRs. Approximate RPFR-values calculated bythe use of the additivity agreed with exact RPFR-values within a margin of 1% error. This error wasequivalent to a 5% error on ln(RPFR). The equilibrium constants of mono boron isotope exhange reactions between three-coordinate boron and four-coordinate boron ranged f r o m 1.0203 to 1.0360 at 25 C,indicating the importance of exact evaluation of RPFRs of polymers.Key words: Ab initio Molecular Orbital Calculations; Polyborates; ReducedRatios; Boron Isotope Exchange; Isotope Fractionation.1. IntroductionThe boron isotope exchange reaction between the neutral boric acid molecule (B(OH)3) and the monomericborate anion (B(OH)4),10B(OH)3 "B(OH)4 "B(OH) 3 10B(OH)4,(1)is a reaction of great concern in boron isotope geochemistry as well as in chromatographic boron isotope separation processes because many boron isotope fractionations observed in those fields are based on it [1], Theequilibrium constant, KB, of Reaction (1) was theoretically obtained as the ratio of the reduced partition function ratios (RPFRs) [2] of the two boron species whichwere calculated through the vibrational analysis in whichforce fields of the two species were constructed so thatthe experimentally observed spectroscopic data were bestreproduced. The /C"B-value was 1.0194 at 25 C and hasbeen the basis for elucidation of observed boron isotopefractionations [1].B(OH) 3 and B(OH)4 are practically the only viableboron species in aqueous solutions of very low boronconcentrations. When the boron concentration is0.025 M or higher, polymeric boron species such asF B Og- (B 3 0 3 (0H)5 2 ) are formed and their existencecannot be ignored any more [3]. The estimation of theRPFRs of such polymeric species is certainly indispensable for better understanding of boron isotope effects insystems containing them. The vibrational analysis ofsuch species, however, has not yet been successful dueto the scantiness of the information on their molecularPartitionFunctionvibrational frequencies. Instead, their RPFR-values wereapproximately estimated from those of the monomericspecies by the use of additivity of the logarithms of theRPFRs (ln(RPFR)s) [1,4]. For example, since H 5 B 3 0 is composed of a triangular group containing a threecoordinate boron atom and two tetrahedral groups, eachof which contains a four-coordinate boron atom, joinedby three common oxygen atoms, its RPFR, (s/s')/H5B3082- w a s approximated asIn (s/s)/H5B3082- {In (s/s ' ) / b ( o h ) 3 2ln(j/s')/B(OH)4-}/3.(2)Generally, when a polyborate contains m three-coordinate boron atoms and n four-coordinate boron atoms, itsRPFR may be approximated asIn (S/S') /polyborate im(s/s')/ B (OH)3 n \x\(s/s')fB{OH) }/(m n). (3)However, the degree of the goodness of this kind ofapproximation has not yet been verified experimentallyor theoreticallyWith the rapid progress in computer technology it isnow possible to carry out ab initio molecular orbital (MO)calculations of vibrational frequencies of polymeric boron species even with a personal computer in relativelyshort times. This enables one to verify the additivity ofln(RPFR)s. In a previous paper [5], the author reportedthe results of the geometry optimization and frequencycalculations at the optimized structures for monomericand dimeric boric acids and borate anions. The RPFRsof the dimers estimated from the RPFRs of the monomers0 9 3 2 - 0 7 8 4 / 00 / 0600-637 06.00 Verlag der Zeitschrift f ü r Naturforschung, Tübingen www.znaturforsch.comDieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschungin Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung derWissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht:Creative Commons Namensnennung 4.0 Lizenz.This work has been digitalized and published in 2013 by Verlag Zeitschriftfür Naturforschung in cooperation with the Max Planck Society for theAdvancement of Science under a Creative Commons Attribution4.0 International License.
624T. Oi Reduced Partition Function Ratios of Boron Species by MO Theoryby the use of the additivity of ln(RPFR)s agreed withthose calculated by using the calculated frequencies ofthe dimers within a margin of 1 %. This error correspondsto an error of 5% of In (RPFR). In this paper, calculationsare extended up to a hexaborate anion and the validity ofApproximation (3) is discussed.2. Theory and Procedure of Calculations2.7. Reduced Partition FunctionRatiosIsotope effects based on molecular translational, rotational and vibrational motions can be estimated by calculating the RPFRs of the chemical species participating inthe considered isotope exchange reaction [2], That reaction may be expressed, without losing any generality, asAX BX' AX' BX,(4)where X and X' are the heavier and lighter isotopes ofthe element considered and A and B are polyatomicgroups. The equilibrium constant, K, of Reaction (4)(strictly speaking, the equilibrium constant estimatedquantum mechanically divided by that estimated classically) can be given as\nK \n(s/s')fBX- ln (s/s')fAX,(5)where (sA') fAX and ln(sA')./Bx a r e the RPFRs of thechemical species AX and BX, respectively. The generalformula of the RPFR of a species is given, under the BornOppenheimer and harmonic oscillator approximations, as(s/s') f TT " / e x PUj/2)/{\ - exp ( - Uj)}i //,'exp ( - u-/2)/{\ - exp ( - u-)}(6)whereUj hc(Oj/(kT),(7)u/ he ü),7 (kT).(7)andHere,/is the degree of freedom of the vibrational motion.h Planck's constant, c the velocity of light, a , and co/the wavenumbers of the ith molecular vibration of theheavier and lighter isotopic species, respectively, k Boltzmann's constant, and T the temperature.package. Based on the results of a previous paper [5], theused ab initio MO theory and basis set were restricted tothe Hartree-Fock self-consistent field (HF) method theory and the 6-31G(d) polarized basis set, respectively. Amore advanced MO theory with a higher level basis setdid not always yield better results of the RPFRs.As polyboric acids and polyborate anions, threetrimers, H 3 B 3 0 6 (B 3 0 3 (0H) 3 ), H4B3O7 (B 3 0 3 (0H)4)and H 5 B 3 0 - ( B 3 0 3 ( 0 H ) ) , one tetramer, H 4 B 4 0 (B 4 0 5 (0H)4 ), one pentamer, H 4 B 5 Or 0 (B 5 0 6 (OH ),and one hexamer, H 6 B 6 0, 2 (B 6 0 6 0(0H)6 ) were considered. The neutral H 3 B 3 0 6 is composed of a 6-membered ring of three B 0 3 triangles and is found in the orthorhombic form of metaboric acid [6]. The H 4 B 3 07 trimerconsists of a ring of two triangles and a B 0 4 tetrahedronand occurs as an anion unit in ameghinite [7], The thirdtrimer B 3 0 3 (0H)5 , consisting of a ring of a triangle andtwo tetrahedra, is found in such boron minerals as inderborite [8]. The tetramer H 4 B 4 09 consists of two 6-membered rings, each of which being composed of a triangleand two tetrahedra with sharing of the two tetrahedra. Itis a borate anion unit in borax [9], The pentamer H 4 B 5 Of 0consists of two 6-membered rings of two triangles and atetrahedron sharing the tetrahedron, and is the unit insborgite [10]. The hexamer H 6 B 6 0 3 consists of three6-membered rings, each of which consisting of a triangle and two tetrahedra with each tetrahedron participating in the formation of two rings. The unique feature ofthis hexamer is the existence of a three-coordinate oxygen atom shared by all the three rings (cf. 0 1 in Fig. If).The H 6 B 6 0 3 unit is found in nobleite [11],The geometries of the polymers were optimized andtheir vibrational frequencies were calculated at the optimized geometries. Calculated frequencies were corrected using the scale factor ( 0.9590) determined for themonomers [5]. With the corrected frequencies, theRPFRs of the polymers were computed using eq. (6), andsubsequently equilibrium constants of various boron isotope exchange reactions were calculated using eq. (5).In all the frequency calculations, the isotopes of hydrogen and oxygen were 'H and 1 6 0, respectively, and onlythe mono boron isotope substitutions were consideredwith the n B basis.3. Results and Discussion2.2. ab initio Molecular OrbitalCalculationsAll the ab initio MO calculations were made with aDell personal computer using the Gaussian 94W programThe most stable structures obtained are depicted inFig. 1; a) for H 3 B 3 0 6 , b) for H 4 B 3 07, c-1) for H 5 B 3 0 ,d) for H 4 B 4 0 ; h e-1) for H 4 B 5 0r ( ) and f) for H 6 B 6 0 ? 3 .
625 T. Oi R e d u c e d Partition Function Ratios of Boron Species by M O TheoryFig. 1. Optimized structures of the polyboric acids and polyborate anions. T h elargest spheres represent B atoms and the intermediate and the small ones presentO and H atoms, respectively. N o significance is attached to the relative sizesof these spheres, a) H 3 B 3 0 6 with C 3 h symmetry, b) H4B3O7 with C 2 symmetry,c-1) H 5 B 3 0X of the lowest energy with C, symmetry, c-2) higher energy conformer of H 5 B 3 0X with Ci symmetry, d) H4B4O9"" with C 2 symmetry, e-1)H 4 B 5 07O of the lowest energy with S 4 symmetry, e-2) higher energy c o n f o r m e r ofH 4 B 5 OJo with S 4 symmetry, f) H 6 H 6 0 f 3 with C 3 symmetry.The structures of the trimers, H 3 B 3 0 6 , H4B3O7 andH 5 B 3 08 , were optimized, assuming C 3h , C2, and C\symmetry, respectively, according to the results by Zhanget al. [12]. The structures of the tetramer, pentamer andhexamer were optimized assuming C 2 , S4 and C 3 symmetry, respectively.For each of the polymers, the geometry optimizationwas also conducted assuming an other symmetry thanthe above one and/or starting from different initial inputdata. Sometimes, the calculation did not converge. Insome cases, the geometry was optimized but negative frequencies were obtained for that structure. In some other
626T. Oi Reduced Partition Function Ratios of Boron Species by M O T h e o r yTable 1. R P F R s of the m o n o m e r s and polymers at 25 C at the HF/6-31G(d) level and additivity of RPFRs.SpeciesFig.B(OH) 3B(OH)IH4B905H5B2O Triangular:TetrahedralSite ofisotopesubstitutionExact.RPFR: ahedraltriangulartetrahedral 0tetrahedral )lb)0:23:02:1H 5 B 3 0X lc-1)1:2H5B3O8(conformer)lc-2)H 4 B 4 0YId)2:2H 4 B 5 07Ole-1)4: 1H4B5OIO(conformer)H 6 B 6 0FRle-2)a3:3If)(B - A) x 100/A.b(B' - A ' ) x 1 0 0 / A ' . c B1 in Fig. lc-1).dApproximateRPFR: ror [5][5][5]B2 in Fig. lc-1). e B l in Fig. l c - 2 ) . 1 B2 in Fig. lc-2).cases, the geometry was optimized and no negative frequency was calculated, but the energy of that structurewas higher than the energy of the structure stated in theprevious paragraph. Thus, the structures in Figs. 1 a), 1 b),lc-1), Id), 1 e-1) and 1 f) are quite probably, although notone hundred percent sure, at the global minima of thepotential energy surfaces.For the H 5 B 3 0 and H 4 B 3 Of 0 borate anions, conformers with higher energies (i.e., structures at local minima)depicted, respectively, in Figs, lc-2) and le-2) are alsoconsidered in order to examine the effect of the structural change on the RPFR. The largest structural differencebetween Figs, lc-1) and lc-2) is that HI is at the eis position to 0 1 in Fig. lc-1), while HI is eis to 0 2 in Fig.1 c-2). The energy of the c-2) structure is higher than thatof c-1) only by 0.18 kJ/mol. The e-2) structure of the pentamer is higher in energy than the e-1) structure by28.5 kJ/mol, about the energy of a typical hydrogen bond.The values of RPFRs of the polymers at 25 C in thiswork as well as those of monomers and dimers in the previous paper [5] are summarized in Table 1. The RPFRsfor mono boron isotope substitutions calculated using (6)are listed as "exact" RPFRs in the fifth column of Table 1.It is observed here that, similarly to the case of themonomers, the values of the RPFRs of the polymers containing up to six boron atoms are smaller for the boronisotope substitutions at the tetrahedral sites than those atthe triangular sites, showing that 1 'B prefers the triangular site to the tetrahedral site in those polymers. This maybe generalized to include polyboric acids and polyborateanions of any size.The RPFR-values of the polymers approximated by(3) using those of the monomers are listed as "approximate" RPFR in the sixth column of Table 1. They agreequite well with the "exact" values within 1 % errors (seventh column). This seemingly shows that the RPFRs ofthe polymers are well approximated by using additivityof the ln(RPFR)s (3). However, this should not be takenliterally, because not the RPFR itself but the deviation ofthe RPFR from unity is physically important, as has beenpointed out in [5]. The goodness of Approximation (3)should thus be estimated as the error of (RPFR - 1 ) or,since the value of the RPFR is usually close to unity, onln(RPFR), instead of RPFR itself. Exact and approximateln(RPFR)-values are listed in the eighth and ninth columns of Table 1, respectively. The percent errors of theapproximate ln(RPFR)-values are listed in the tenth column. A maximum deviation of about 5% is observed. One
627 T. Oi Reduced Partition Function Ratios of Boron Species by M O TheoryTable 2. Equilibrium constants (K) of mono boron isotope exchanges between various boric acid and borate monomers and polymers at 25 C a b : n X l() Y l ( ) X " Y K ([' X] [ " Y ] ) / ( [ n X ] [ l 0 Y]).B(OH) 3H4B2O5H5B2OFIH3B3O6H 4 B 3 0YH5B3OJ-H4B4o2"H4B5O10H 6 B 6 O, 2 3 -B(OH) 41.0260H 5 B 2 -34.0)1.0235( (-24.6)1.0207( 20.2)1.0274(-5.3)1.0307(-17.8)1.0247( 9.5)1.0297(-14.0)1.0222( 2(-19.7)1.0236( (-25.0)1.0206( 20.6)1.0273(-4.9)1.0306(-17.4)1.0246( .1)1.0296(-13.6)1.0203( 21.7)1.0270(-3.8)1.0304(-16.7)1.0243( .0)1.0293(-12.5)1.0238( (-25.7)1.0225( 5(-20.8)Y(triangularsite)X(tetrahedral site)X H 6 B 2 0Y"H4B3O7H 5 B 3 0X H5B3OX2"DH 4 B 4 0YH4B5OT0H 6 B 6 0, 2 3athe upper row; the K value. h the lower row (in the parenthesis); percent deviation of the lnA -value of Reaction (1) from theexact value. c B1 in Fig. lc-1). d B2 in Fig. lc-1).may expect similar levels of deviations for larger polymers. The judgement on whether this deviation is acceptable or too large will depend on the accuracy of theapproximation one requires.RPFR-values of higherenergy conformers of H5B3Og and H 4 B 5 Of 0 are also listed in Table 1. For both the polymers, the RPFRs of the more stable structures (Figs.1 c-1 and 1 e-1) are larger than those of the less stable ones(Figs, lc-2 and le-2) for substitutions both at tetrahedraland triangular sites. Although it is not certain whetherthis observation can be generalized, these results are reasonable and understandable; the curvature at the bottomof a deeper potential energy curve is usually larger thanthat of a shallower potential energy curve.The equilibrium constants (AT) of the boron isotopeexchange reactions between boron in a B 0 3 triangle ofone species and boron in a B 0 4 tetrahedron of another at25 C are summarized in Table 2. The l v a l u e rangesfrom 1.0203 for the boron isotope exchange reaction ofnB ( O H ) i H 4 l ( ) B 1 1 B 3 0;;- ,0B(OH)i H 4 l l B 4 O to 1.0360 for the reaction ofH6"B20 H61010B(OH) 3B " B O "B(OH)3.One can expect that the range of the AT-value will not beenlarged substantially even if larger polymers are included in the tabulation. All the exchange reactions of Table 2do not occur in real systems. Table 2 thus gives a roughestimate on how polymer formations can affect the boronisotope fractionation in a system of concern. In the parentheses shown are the percent deviations from the lnA'gvalue for Reaction (1), which are nothing but the percenterrors obtained when Approximation (3) is used for RPFRcalculations of polymers. A largest deviation of about 40%is observed in Table 2, indicating that this much deviationcan occur as a result of polymer formations.As mentioned earlier, the vibrational analysis gave a/C -value of 1.0194 at 25 C. Experimentally, some conflicting results were presented concerning the A -value.In the field of boron isotope geochemistry, boron isotopefractionations between seawater and natural carbonatesreported by Hemming and Hanson [13] and by Gaillardet and Allegre [14] were consistent with the A'g-valueof 1.0194. Contrary to these, larger A -values wererequired to explain the boron isotope effects observed byVengosh et al. [15] and by Palmer et al. [16]. For example, the results of Palmeret al. [16] indicated K B 1.033.In the field of boron isotope separation in aqueoussystems, results indicating that the A -value should belarger than 1.0194 were also reported [17]. According to
628T. Oi Reduced Partition Function Ratios of Boron Species by MO Theorythe results of Table 2, the existence of polymeric boronspecies may elucidate the variation in boron isotope fractionation obtained experimentally.4. ConclusionsTo summarize, the following statements can be made:Geometries of polyboric acids and polyborate anions upto a hexamer were optimized at the HF/6-31G(d) level,and their RPFRs were evaluated exactly and approximately. Approximate RPFR-values by (3) agreed withexact RPFR-values within a margin of 1 % error, whichcorresponds to a 5% error in ln(RPFR). The K values[1] H. Kakihana, M. Kotaka, S. Satoh. M. Nomura, and M.Okamoto. Bull. Chem. Soc. Japan 50,158 (1977).[2] J. Biseleisen and M. G. Mayer. J. Chem. Phys. 15, 261(1947).[3] N. Ingri, Svensk Kemisk Tidskift 75, 199 (1963).[4] T. Oi, M. Nomura, M. Musashi. T. Ossaka, M. Okamoto,and H. Kakihana. Geochim. Cosmochim. Acta 53, 3189(1989).[5] T. Oi, J. Nucl. Sei. Technol. 37, 166 (2000).[6] C. R. Peters and M. E. Milberg, Acta Cryst. 17,229 (1964).[7] A. Dal Negro, J. M. M. Pozas, and L. Ungaretti, Amer.Mineral. 60, 879 (1975).[8] E. N. Kurkutova, I. M. Rumanova, and N. V. Belov, Sov.Phys. Dokl. 10, 808 (1966).[9] N. Morimoto, Mineral. J. (Japan) 2, 1 (1956).[10] S. Marlino and F. Sartori. Acta Cryst. B28, 3559 (1972).between two of various monomeric and dimeric boricacids and borate anions varied from 1.0203 to 1.0360 at25 C. The maximum percent deviation of an approximately evaluated K value from the exact value was about38%, indicating the importance of accurate evaluationsof the RPFRs of polymers. The present work may helpto elucidate boron isotope fractionation obtained experimentally.AcknowledgementThis study was financially supported in part by JapanSociety for the Promotion of Science under the 19982000 Japan-China Scientific Cooperation Program.[11] J. R. Clark, Amer. Mineral. 49, 1549 (1964).[12] Z. G. Zhang, M. B. Boisen, Jr., L. W. Finger, and G. V.Gibbs, Amer. Mineral. 70, 1238 (1985).[13] N. G. Hemmine and G. N. Hanson, Geochim. Cosmochim. Acta 56, 537 (1992).[14] J. Gaillardet and C. J. Alleere, Earth Planet. Sei. Lett. 136,665 (1995).[15] A. Vengosh, Y. Kolodny, A. Starinsky, A. R. Chivas, andM. T. McCulloch, Geochim. Cosmochim. Acta 55, 2901(1991).[16] M. R. Palmer, A. J. Spivack, and J. M. Edmond, Geochim.Cosmochim. Acta 51, 2319 (1987).[17] M. M. Urgell, J. Iglesias, J. Casas, J. M. Saviron, and M.Quintanilla, Third UN Int'l. Conf. on Peaceful Uses ofAtomic Energy, A/CONF.28/P/491. Spain 1964.
2.2. ab initio Molecular Orbital Calculations All the ab initio MO calculations were made with a Dell personal computer using the Gaussian 94W program package. Based on the results of a previous paper [5], the used ab initio MO theory and basis set were restricted to the Hartree-Fock self-consistent field (HF) method the-
referred to as "FMO-MD," is an ab initio MD method (Komeiji et al., 2003) based on FMO, a highly parallelizable ab initio molecular orbital (MO) method (Kitaura et al., 1999). Like any ab initio MD method, FMO-MD can simulate molecular phenomena involving electronic structure changes such as polarization, electron transfer, and reaction.
T3. Ab Initio Hartree-Fock Calculations The point of the empirical parameters in semiempirical calculations was to cut down on the number of electron-electron repulsion integrals that needed to be computed. In ab initio calculations one simply calculates them all. This inevitably means that ab initio calculations take much longer than semiempirical
calculation. The results of ab initio methods can depend on the choice of the initial orbital guess. At every point we are interested in we apply a series of ab initio calculations, where each ab initio calculation uses the orbitals from the previous calculation as initial orbital guess. We start with a spin-restricted Hartree-Fock
Ab-initio Calculations of the Structural, Electronic and Optical Properties of (CdSe) 2 Clusters . The calculated highest occupied molecular orbital (HOMO), and the lowest unoccupied molecular orbital (LUMO) gap of the isomers implied that the gap depends on the symmetrical geometry of the isomer. Furthermore,
Ab initio calculations, which confirmed the experimental findings [9], indicate that this red shift derives from a strong charge transfer to the O-H σ orbital of a single water molecule. This water molecule points both its hydrogen atoms toward the electronic cloud in the so . Ab initio molecular dynamics simulations were performed .
semiclassical force fields to ab initio calculations,33 35 or performing QM/MM calculations with the environment being simulated classically.36 A more direct solution, developed in the current work, employs the FMO molecular dynamics (FMOMD) method37 41 to simplify the computational complexity of ab initio NAMD. The FMO technique is used
Theoretical Atomic & Molecular Calculations. Atomic structure Atomic scattering Molecular Structure Molecular Scattering . . Validating ab initio calculations by state-selective triple-resonance spectroscopy . Hartree-Fock "Orbital approximation" variational 2. Electron correlation configuration interaction variational
reading comprehension. DIRECTIONS. this practice test contains one reading selection with two multiple-choice questions and one open-response question. Mark your answers to these questions in the spaces provided on page 5 of your practice test answer document. 1. The porcupine is a controversial, yet important, forest creature. Our more prickly encounters with “quill pigs” may be remedied .
