Jaynes information entropy of small molecules: Numerical evidence of the Collins conjecture

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1 PHYSICAL REVIEW A VOLUME 56, NUMBER 6 DECEMBER 1997 Jaynes information entropy of small molecules: Numerical evidence of the Collins conjecture Juan Carlos Ramírez, Catalina Soriano, Rodolfo O. Esquivel, and Robin P. Sagar Departamento de Química, Universidad Autónoma Metropolitana, Apartado Postal , Iztapalapa, México, Distrito Federal, Mexico Minhhuy Hô and Vedene H. Smith, Jr. Department of Chemistry, Queen s University, Kingston, Ontario, Canada K7L 3N6 Received 23 December 1996; revised manuscript received 11 July 1997 Collins s conjecture Z. Naturforsch. 48A, that the correlation energy of a system is proportional to the Jaynes entropy in Papers on Probability, Statistics and Statistical Physics, edited by R. Rosencrantz Reidel, Dordrecht, 1983 is investigated for small molecular systems. Numerical evidence supporting the conjecture is obtained by computing entropies from highly correlated configuration-interaction wave functions at various correlation levels, using sequences of increasingly larger basis sets. Further evidence is also obtained by examination of results obtained from a variety of correlation methods utilizing a fixed basis set. S PACS numbers: Jv, p I. INTRODUCTION Information theory 1, which was first utilized in the study of communication circuits 2, has since had a major impact on a number of diverse fields of investigation. In quantum chemistry, the aim of investigators has been in utilizing information theory to characterize the nature of the information content of a chemical system in order to ascertain whether such a theory yields useful concepts for the studies of atomic and molecular systems Through such studies, important conceptual advances in the relationship between information entropies and chemical structures have been advanced. One major challenge lies in the development of physical concepts that may be easily interpreted within an information theoretical perspective. The one-particle reduced density matrix of an N-particle system may be defined as 1;1N 1,2,...,N*1,2,...,Nd2 dn 1.1 in terms of the N-particle wave function, where the index denotes a combined spin and spatial coordinate of a particular electron. Knowledge of this matrix is sufficient to determine all one-electron properties of the system and also twoelectron properties such as the energy, within the Hartree- Fock HF approximation. This matrix may be expanded in a set of orthonormal spin orbitals ı 1;1 i, j c ij i 1 j *1 1.2 and then diagonalized to yield 1;1 j v j j 1 j *1, 1.3 where the eigenfunctions j are the natural spin orbitals and v j are the occupation numbers. One-particle density matrices have occupation numbers that sum to N and lie in the range 0,1 in order to be N representable 11. Also, within the independent-particle model, these occupation numbers are either zero or one, which identifies (1;1) as idempotent. The spin-traced charge density matrix may be defined in terms of the one-particle reduced density matrix as r;r 1;1 s1 s 1 ds 1, 1.4 where s 1 corresponds to the spin coordinate of electron 1, while r refers to the spatial coordinates. There is a similar expansion of (r;r) to Eq. 1.3, namely, r;r i n i i r i *r. 1.5 It is the diagonal element of (r;r) that corresponds to the spin-traced charge density (r), which may be expressed as r i n i i 2, 1.6 where n i are the occupation numbers that lie in the range 0,2 of the natural orbitals i. From (r) and the spin-traced momentum density (p), the Shannon entropies can be defined as S Shannon rlnrdr, S Shannon plnpdp These quantities have been investigated as measures of basisset quality 5, of electron correlation 6, and with regard to geometrical changes /97/566/44776/$ The American Physical Society

2 4478 JUAN CARLOS RAMÍREZ et al. 56 Another avenue of investigation has been an examination of the Jaynes entropy of a spin-traced charge density matrix, defined as S Jaynes Trr;rlnr;r, which may be computed as 1,12 S Jaynes n i lnn i, where the n i s are as defined in Eq Similarly, on (1;1) of Eq. 1.3 is S Jaynes Note that S Jaynes S Jaynes v j lnv j is related to S Jaynes by S Jaynes S Jaynes N ln2 for singlet systems 12. Collins 12 has proposed a physical interpretation of the Jaynes information entropy in that it is proportional to the correlation energy, which may be defined as 0 E corr E 0 E HF In the above, E 0 is the true ground-state energy and E HF is the Hartree-Fock energy, with the correlation energy achieving its maximum value, zero, for an independent-particle system (E 0 E HF ). Thus properties of the Jaynes information entropy are properties of the correlation energy E corr k n i lnn i, 1.13 where k is a constant. For the charge density matrix (r;r), the entropy of an independent-particle state S RHF may be defined as S close RHF N2 ln2n ln for closed-shell systems, where N is the number of doubly occupied orbitals. For open-shell systems, one has to take into account that the contribution from a singly occupied orbital (1 ln1) is zero and thus the summation has to be adjusted accordingly. For the one-particle density matrix (1;1), the eigenvalues v i of an independent-particle state are either zero or one from Eq Thus the Jaynes entropy from Eq for such a state would be zero. Since the correlation energy of an independent-particle state is conventionally zero, Eq may be rewritten for a closed- or an open-shell system as similar use of the nonidempotency of the one-particle density matrix as a measure of correlation strength has been proposed 14,15. Collins s conjecture has been investigated recently 16 at the atomic level for the lithium atom isoelectronic sequence. The results that were obtained strongly supported the conjecture. Most notably, a linear relationship between the entropy and the correlation energy was observed for the individual members of the lithium sequence. On the practical side, it was suggested 16 that the Jaynes entropy could be used as a measure of basis-set quality within a given correlation model e.g., full configuration-interaction CI or to characterize the strength of the correlation in a system in a manner similar to that for the correlation energy. In this work we will extend our studies to molecular systems. We will probe the validity of Collins s conjecture for small 10-, 14-, and 18-electron molecules using different correlation methods and a variety of basis sets. These basis sets were used to prepare systems with varying amounts of correlation energy in order to test the hypothesis. We will study the validity of Eq since the value of C in Eq has the effect of shifting values, i.e., setting the righthand side of Eq equal to zero for an independentparticle state, and does not affect the linearity. II. RESULTS AND DISCUSSION Molecular wave functions were calculated at the CI level for isoelectronic groups of 10-, 14-, and 18-electron systems. For the 10-electron systems we studied HF, H 2 O, CH 4, and NH 3. For the 14-electron systems CO, HCCH, HCN and N 2, while for the 18-electron case CH 3 -CH 3, CH 3 OH, H 2 S, CH 3 -NH 2,CH 3 -F, H 2 O 2 and HCl were studied. These molecules were chosen as representative of the different isoelectronic groups. The calculations were performed at the CI with single and double excitation CISD and quadratic CISD QCISD levels using the GAUSSIAN94 17 suite of programs, utilizing eight different basis sets, STO-3G, 3-21G, 6-31G*, 6-31G**, 6-31G*, 6-31G**, 6-311G**, and 6-311G** using standard notations of Ref. 17 where E corr k n i lnn i C, 1.15 where CkS RHF from the above equations. Equation 1.15 shows a measure of the deviation of the occupation numbers of a correlated system, from the Hartree-Fock values, from an entropy point of view. We also note that a FIG. 1. Number of basis functions used in the CISD calculation versus correlation energy dashed line and Jaynes entropy solid line, for the NH 3 molecule. Energies and entropies are given in atomic units.

3 56 JAYNES INFORMATION ENTROPY OF SMALL TABLE I. Behavior of the entropies with respect to the correlation energies for the molecules studied: All indicates that all basis sets comply with the expected monotonic behavior, while numbers in parentheses indicate the number of exceptions and the corresponding basis sets. Molecule CISD QCISD CH 4 All All NH 3 All All H 2 O G**, 6311G**, 6311G** G** HF G**, 6311G** G**, 6311G** HCCH All All HCN All 1 631G* N G* All CO G* All H 2 S All All HCl All All CH 3 CH 3 All All CH 3 F All All CH 3 NH 2 All All CH 3 OH All All HOOH 3 631G*, 6311G**, 6311G** 3 631G*, 6311G**, 6311G** STO denotes Slater-type orbitals and G denotes Gaussiantype orbitals. These basis sets were chosen in order to generate wave functions with varying amounts of correlation energy. For the cases of N 2 and CO, five basis sets STO-3G, 3-21G, 6-31G*, 6-31G*, and 6-311G* were studied since the absence of hydrogen excludes basis sets with double polarization and double diffuse functions. For the H 2 S and HCl molecules, the basis sets 6-311G** and 6-311G** were omitted for the sake of consistency, since for the second-row atoms these basis sets employ the McLean-Chandler 18 contraction scheme, which differs significantly from the 6-31 basis set. S Jaynes was calculated from Eq The correlation energy for a particular basis set and correlation method CM was calculated from E corr E CM E RHF 2.1 using the restricted HF RHF and CM results of a particular basis set. All calculations were performed on a Silicon Graphics Indigo 2 computer. We present in Fig. 1 a plot of the number of contracted basis functions basis sets used in the CISD calculation versus S Jaynes and E corr, respectively, for the NH 3 molecule. This plot is representative of what was observed for the majority of the studied molecules. In such a manner, we may compare the behavior of the correlation energy, and through the entropy, the behavior of the charge density matrix. One observes that the energy decreases as the number of contracted basis functions is increased or the quality of the basis set is improved, while the entropy increases, i.e., the entropy behaves in an opposite sense to the energy. Thus the correlation energy is monotonically decreasing as a function of the basis-set size. The exceptions to this behavior for the correlation energy occurred on going from the 6-31G** to the 6-31G* basis sets, in certain molecules with hydrogen HCCH, HCN, HCl, and HOOH, where it is important to include the polarization functions for a correct description of other properties of the system. Thus the addition of more basis functions does not necessarily yield a better correlation energy. When the energy oscillates, so does the entropy in the majority of the studied cases. It is also apparent from Fig. TABLE II. Values of the Jaynes entropy and total energies for different basis sets of the CH 3 OH molecule at the CISD and QCISD levels. The Hartree-Fock energies of the basis sets are also included. All values are in atomic units. CISD QCISD Basis S Jaynes Energy S Jaynes Energy Hartree-Fock energy STO-3G G G* G** G* G** G** (7) (66) (25) (950) 6-311G** (2) (68) (27) (957)

4 4480 JUAN CARLOS RAMÍREZ et al. 56 FIG. 2. Plot of the Jaynes entropy versus the correlation energy for the 10-electron systems at the CISD level. Triangles correspond to HF, diamonds to H 2 O, squares with the dashed line to NH 3, and squares with the solid line to CH 4. Energies and entropies are given in atomic units. Note that the points corresponding to the eighth basis set lie within the radius of the symbols belonging to the seventh basis set in the lower right of the plot. 1 that overall the Jaynes entropy is a monotonically increasing function of the basis-set size. Moreover, the trends present in Fig. 1 were also indicative of the behavior that was observed for the QCISD level. In Table I we summarize for the other molecules the data that were presented for NH 3 in Fig. 1. One can see that for ten out of the fifteen molecules CISD and for eleven out of fifteen molecules QCISD, the behavior presented in Fig. 1 holds. Of these exceptions, the worst case occurred when three basis sets out of the studied eight did not reflect the monotonic relationship between the energy and the entropy. These results present numerical evidence that the entropy behaves in the opposite sense to the energy. Moreover, for eleven out of the fifteen molecules studied, the behavior at the QCISD level was observed to be the same as at the CISD level. In Table II we report values of S Jaynes and total energies of the CH 3 OH molecule for basis sets at the CISD and QCISD levels. These values are representative of what was observed for most of the other cases. One may observe a decrease in energy on going from the CISD to the QCISD levels the only exception occurred for the STO-3G basis of the HF and HCl molecules, which gave the same energy at both levels. Also, one can see that S QCISD Jaynes is always greater than S CISD Jaynes. This was observed to be true for all the molecules studied with the exception of the STO-3G for HF and HCl. These values for the entropy present evidence that it is sensitive to the quality of the wave function and may be used as a measure of basis-set quality. Such results may also be interpreted physically. As we improve upon the correlation model, electrons are able to avoid each other more easily, therefore, the electron distribution becomes more spread out. Thus one would expect that the entropy for such a distribution would tend to a maximum in accordance with the maximum-entropy principle 1, which states that the leastbiased distribution maximum spread possesses a maximum in entropy. Examples of the sensitivity of the Jaynes entropy to the correlation energy are presented for the 10-electron systems in Fig. 2. The results show that the Jaynes entropy is proportional to the correlation energy. A linear fit of the data y E corr, xs Jaynes for both CISD and QCISD levels yielded correlation coefficients that were suggestive of a linear relationship i.e., in the range for the majority. The results of the fits for all the molecules studied have been compiled and presented in Table III. First, one can observe a great similarity of slopes in all but the two molecules that contain second-row atoms. The TABLE III. Statistical data slopes, values of x(0), and correlation coefficients obtained from linear fits of the correlation energy to the Jaynes entropy, using different basis sets, for 10-, 14-, and 18-electron systems. CISD QCISD Molecule Slope x(0) R Slope x(0) R CH NH H 2 O HF HCCH HCN N CO H 2 S HCl CH 3 CH CH 3 F CH 3 NH CH 3 OH HOOH

5 56 JAYNES INFORMATION ENTROPY OF SMALL TABLE IV. Number of configuration-state functions, Jaynes entropies, and energies for the H 2 O molecule calculated using the D95 basis set 20 in a variety of correlation models. All values are in atomic units. Method CSF S Jaynes Total energy E corr RHF CASSCF CASSCF CASSCF CASSCF CISD CISDT (1) MRCISD MRCISD (9) MRCISD (5) CISDTQ (6) MRCISD (8) slopes for the latter two are similar to each other. Note that the slopes in these fits correspond to the constant k, in Eq Such a similarity, independent of the size of the system, suggests that there might be a unique constant k by which the correlation energy may be proportioned to the entropy. Except for HF, the CISD slopes are all more negative than the QCISD ones. Second, the x(0) values where the fits cross the x axis a zero of correlation energy are also presented. These values correspond to an independent-particle state RHF and from 10 Eqs and 1.15 should be equal to S RHF for ten electrons, S RHF for fourteen electrons, and 18 S RHF for eighteen electrons. One can see from the reported values that there is a definite trend towards these values for the members of each group, with HF, N 2, and HCl being the closest from their respective groups. Third, a comparison of CISD and QCISD values indicates that the relationship between the Jaynes entropy and the correlation energy holds, independent of the correlation model used. To further substantiate this, we have also studied the behavior of the Jaynes entropy versus several correlation models for a fixed basis set. We chose the H 2 O molecule since there is a wealth of information readily available for comparison. The geometry used was identical to that previously reported 19. The D95 basis set used for the calculation is Dunning s contraction 20 of Huzinaga s primitive O(9s5p), H(4s) basis 21. The three correlation models were chosen as the CI, complete active space self-consistent field CASSCF, and multireference CISD MRCISD methods. The number of configuration-state functions CSF s available from these models was then used for comparison to the corresponding energies and entropies in a manner analogous to what was done for the number of basis functions. In the CASSCF model, the ten electrons are distributed in an increasing number of six, seven, eight, and nine active orbitals. These active orbitals were added in the order of increasing self-consistent-field orbital energies. For the MRCI model, the corresponding CASSCF s were used as the references for the subsequent CISD calculations. The CI model involving CISD, CISD with triple excitations TABLE V. Statistical data slopes, values of x(0), and correlation coefficients obtained from linear fits of the correlation energy to the Jaynes entropy, using different correlation methods and the D95 20 basis set for H 2 O. Method Slope x(0) R All points CASSCF MRCISD CI CISDT, and CISDT with quadruple excitations CISDTQ were treated in the conventional way. All of these calculations were performed using the HONDO 8.5 program 22. Table IV reports the number of configuration-state functions, Jaynes entropies, total energies, and correlation energies. The results for the energies are in agreement with those previously reported 23. An examination of the relationship between the CSF s and E corr reveals that one consistently recovers more correlation energy as one expands the configuration space, within a particular model. However, this gain is not as obvious in going from one correlation model to another. For example, the number of CSF s in the CASSCF-9 method is much greater than the CISD case; however, the amount of correlation energy recovered by CASSCF-9 is not as great as that of CISD. It is believed that the CASSCF method forfeits the energy advantage for the proper dissociation behavior 23. This same type of argument may be used to explain the difference between MRCISD-8 and CISDTQ. There is a correspondence between S Jaynes and E corr as may be observed throughout the entire series, i.e., the entropy increases, while the correlation energy decreases. The only exceptions occur at the CISD and CISDTQ values where there is a jump to a different correlation model as documented in the preceding paragraph. It is striking that there is a smooth relationship within a particular method where the correlation energy and occupation numbers hence Jaynes entropy behave in a systematic and predictable fashion. These values support the Collins conjecture. At the same time, the kinks observed in Table IV further quantify the subtle differences in the wave functions that manifest themselves through the occupation number formalism of the Jaynes entropy. We have performed linear fits of the data points in Table IV and presented the results in Table V. One can observe that the fits conform more to linearity correlation coefficients closer to one when the methods are treated separately. Also, the values of the slopes and x(0) are similar in sign and order of magnitude for the three different methods that were studied, with the CASSCF method providing a value of x(0) closest to the theoretical value. III. CONCLUSION Configuration-interaction molecular wave functions for 10-, 14-, and 18-electron systems were calculated at the CISD and QCISD levels, using a variety of different basis sets in order to create wave functions with varying degrees of correlation energy. The Jaynes entropy was then calculated

6 4482 JUAN CARLOS RAMÍREZ et al. 56 from these wave functions and the data were used to study the relationship between the correlation energy and the Jaynes entropy. The Jaynes entropy at the QCISD level was found to be greater than that at the CISD level, while the correlation energies at the QCISD level were found to be lower than at the CISD. In all of the molecules studied, the numerical evidence suggests a strong relationship between the Jaynes entropy and the correlation energy, with 17 exceptions out of 220 data points. Furthermore, the monotonic relationship between decreasing correlation energy and increasing entropy was also observed among a variety of different correlation methods involving a fixed basis set. These results add evidence in support of Collins s conjecture at the molecular level and provide a physical foundation for the entropy. It would be interesting to study the behavior of the Jaynes entropy with regard to conformational changes in a molecule and also in a chemical reaction. ACKNOWLEDGMENTS This research was supported in part by the Natural Sciences and Engineering Research Council of Canada. R.O.E. wishes to acknowledge financial support from the Consejo Nacional de Ciencia y Tecnología México through Grant No E9406. Allocation of computer time from the Laboratorio de Supercómputo y Visualización en Paralelo at the Universidad Autónoma Metropolitana-Iztapalapa is gratefully acknowledged. 1 E. Jaynes, in Papers on Probability, Statistics and Statistical Physics, edited by R. Rosencrantz Reidel, Dordrecht, C. E. Shannon, Bell. Syst. Tech. J. 27, S. R. Gadre, S. B. Sears, S. J. Chakravorty, and R. D. Bendale, Phys. Rev. A 32, I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, M. Hô, R. P. Sagar, J. M. Pérez-Jordá, V. H. Smith, Jr., and R. O. Esquivel, Chem. Phys. Lett. 219, M. Hô, R. P. Sagar, R. O. Esquivel, and V. H. Smith, Jr., J. Phys. B 27, S. B. Sears, R. G. Parr, and U. Dinur, Int. J. Quantum Chem. 19, M. Hô, H. Schmider, R. P. Sagar, D. F. Weaver, and V. H. Smith, Jr., Int. J. Quantum Chem. 53, M. Hô, R. P. Sagar, D. F. Weaver, and V. H. Smith, Jr., Int. J. Quantum Chem. 29, A. Nagy and R. G. Parr, Int. J. Quantum Chem. 58, , and references therein. 11 P. O. Lowdin, Phys. Rev. 97, ; 97, D. M. Collins, Z. Naturforsch. 48A, Note that the Jaynes s entropies on the two representations (r;r) and (p;p) are the same. 14 P. Ziesche, Int. J. Quantum Chem. 56, P. Gersdorf, W. John, J. P. Perdew, and P. Ziesche, Int. J. Quantum Chem. 61, R. O. Esquivel, A. L. Rodríguez, M. Hô, R. P. Sagar, and V. H. Smith, Jr., Phys. Rev. A 54, M. J. Frisch, G. W. Trucks, M. Head-Gordon, P. M. W. Gill, M. W. Wong, J. B. Foresman, B. G. Johnson, H. B. Schlegel, M. A. Robb, E. S. Replogle, R. Gomperts, J. L. Andres, K. Raghavachari, J. S. Binkley, C. Gonzalez, R. L. Martin, D. J. Fox, D. J. DeFrees, J. Baker, J. J. P. Stewart, and J. A. Pople, GAUSSIAN94, Gaussian Inc., Pittsburgh, PA, A. D. McLean and G. S. Chandler, J. Chem. Phys. 72, P. Saxe, H. F. Schaefer III, and N. C. Handy, Chem. Phys. Lett. 79, T. H. Dunning, J. Chem. Phys. 53, S. Huzinaga, J. Chem. Phys. 42, HONDO 8.5 from CHEM-Station, 1994, IBM Corporation, Neighborhood Road, Kingston, NY F. B. Brown, I. Shavitt, and R. Sheppard, Chem. Phys. Lett. 105,