Hongo  Kenta

Kawazoe  Yoshiyuki

Maruyama  Youhei

Oyamada  Takayuki

Yasuhara  Hiroshi

MATERIALS TRANSACTIONS

Hongo, Kenta

103313

Oyamada, Takayuki

103314

Maruyama, Youhei

103315

Kawazoe, Yoshiyuki

103316

Yasuhara, Hiroshi

103317

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

Correct Interpretation of Hund's Rule andChemical Bonding Based on the Virial Theorem著者 Hongo  Kenta, Oyamada  Takayuki, Maruyama Youhei, Kawazoe  Yoshiyuki, Yasuhara  Hiroshijournal orpublication titleMaterials Transactionsvolume 48number 4page range 662-665year 2007URL http://hdl.handle.net/10097/52284Correct Interpretation of Hund’s Rule and Chemical BondingBased on the Virial TheoremKenta Hongo, Takayuki Oyamada, Youhei Maruyama,Yoshiyuki Kawazoe and Hiroshi YasuharaInstitute for Materials Research, Tohoku University, Sendai 980-8577, JapanWe have investigated Hund’s spin-multiplicity rule for the second and third row atoms (C, N, O, Si, P, and S) and the methylene molecule(CH2) by means of diffusion Monte Carlo method and complete active space self-consistent field method, respectively. It is found that Hund’srule is interpreted to be ascribed to a lowering in the electron-nucleus attractive Coulomb interaction energy which is realized at the cost ofincreasing the electron-electron repulsive Coulomb interaction energy as well as the kinetic energy. We have also studied correlation in thehydrogen molecule H2. Correlation in H2 gives an increase of the electron density distribution nðrÞ in the left and right anti-binding regions, areduction of nðrÞ in the binding region, and an increase in the equilibrium internuclear separation. The importance of the virial theorem isstressed in the evaluation of correlation effects on both Hund’s rule and chemical bonding in H2. [doi:10.2320/matertrans.48.662](Received November 6, 2006; Accepted February 15, 2007; Published March 25, 2007)Keywords: virial theorem, Hund’s rule, chemical bonding, electron correlation1. IntroductionThe virial theorem 2T þ V ¼ 0 holds for any stationarystate of many-electron systems,1–3) where T is the kineticenergy and Vð¼ Ven þ Vee þ VnnÞ the total Coulomb poten-tial energy; Ven, Vee, and Vnn are the electron-nucleus, theelectron-electron, and the nucleus-nucleus Coulomb poten-tial energies, respectively. Hence the theorem is a necessarycondition for an accurate evaluation of the stationary stateenergies of atoms, molecules and solids as well as theirvarious electronic properties. In accordance with the virialtheorem we have given for the first time the correctinterpretation of Hund’s spin-multiplicity rule for six atomsand a molecule by diffusion Monte Carlo (DMC) method andcomplete active space self-consistent field (CASSCF) meth-od, respectively. We have also studied how correlationchanges the electron density distribution nðrÞ in the hydrogenmolecule H2.Hund’s rule states that the highest spin-multiplicity state isthe most stable among all the possible states arising from asingle electronic configuration.4,5) This empirical rule isapplied to the ground state of atoms, ions, and molecules andeven their low-lying excited states in most cases. In almost alltextbooks excluding Levine’s,3) Hund’s rule has long beeninterpreted to be ascribed to a reduction in Vee in the groundstate with the highest spin-multiplicity, i.e., the largestpossible amount of exchange energy.6,7) This traditionalinterpretation assumes that the ground and the lowest excitedstates could have the same set of orbitals. It gives the same Tand Ven for the two states and ascribes the energy differenceto Vee. In spite of giving the correct ordering of the two states,the traditional interpretation is invalid since it violates thevirial theorem. Correctly, the two states independently fulfillthe virial theorem and hence 2T þV ¼ 0, i.e., thestabilization of the ground state is realized by lowering Vby an amount of 2T at the cost of increasing T byTðT > 0Þ.A number of authors have studied Hund’s rule foratoms8–19) and light molecules20–28) by Hartree-Fock (HF)and other variational calculations. They have found that thestabilization of the highest multiplicity state relative to thelower multiplicity states is ascribed to a lowering in Ven thatis gained at the cost of increasing Vee as well as T .Davidson8,9) has first pointed out that Vee is larger for thetriplet than for the singlet by HF calculations for low-lyingexcited states of the helium atom. Boyd17) has first introducedthe concept of less screening to explain how Ven is lowered.In the higher multiplicity state there are more parallel spinpairs and hence a larger Fermi hole around each electron.Owing to the presence of a larger Fermi hole the Hartreescreening of the nuclear charge is reduced more effectively atshort interelectronic distances in the higher multiplicity state.Thus, each electron experiences the nuclear charge moreeffectively in the higher multiplicity state and hence theelectron density distribution contracts around the nucleus togive a lowering in Ven and an increase in both T and Vee.The Heitler-London (HL) model29,30) has long beenappreciated as the pioneering study of chemical bonding inH2, but in fact fails to describe the essence of chemicalbonding since it violates the virial theorem.31–33) The linearcombination of 1s atomic orbitals (LCAO) also violates thetheorem. The minimal basis full configuration interaction(CI), i.e., an interpolation of HL and LCAO is the bestapproach to H2 so far as one uses 1s atomic orbitals alone. Anumber of solid-state physicists still believe that the truth ofchemical bonding in H2 lies in between HL and LCAO34–37)since such an interpolation compensates for individualshortcomings. However, the minimal basis full CI is farfrom describing the essence of chemical bonding since itviolates the virial theorem. The truth does not lie in between.The virial theorem holds for both a molecule and itsconstituent atoms. Hence the binding energy of a moleculeEðE < 0Þ is given as E ¼ T ¼ V=2, where T isan increase in T andV a lowering in V involved in chemicalbonding. The essence of chemical bonding is an inevitableincrease in T , T and a concomitant lowering in V , Vwhose magnitude equals 2T .Materials Transactions, Vol. 48, No. 4 (2007) pp. 662 to 665Special Issue on ACCMS Working Group Meeting on Clusters and Nanomaterials#2007 Society of Nano Science and Technology2. Hund’s Rule for the Second and Third Row AtomsFigure 1 shows the energy differences E,T ,Ven, andVee between the highest (high-S) and the next highest (low-S) multiplicity states for the 2p and 3p atoms, evaluated fromHF38) and DMC.39) We have ascertained that the followingrelations hold for any atom calculated: E  Ehigh-S Elow-S < 0, Ven  Vhigh-Sen  V low-Sen < 0, T  Thigh-S T low-S > 0, and Vee  Vhigh-See  V low-See > 0. The stabiliza-tion of high-S state relative to low-S state, or equivalently thenegative value of E is ascribed to that Ven < 0. Theenergy difference Ven is largest in magnitude among all theenergy differences. The lower value of E in the ground stateis a direct consequence of the lower value of Ven therein. Thisproperty may be termed the predominance of Ven in theground state.Correlation gives a rather small change in the energydifference E between the two states for all the atoms,though it lowers the energy level for both high-S and low-Sstates by about 1 eV per electron. On the other hand,correlation increases both Vee and Ven in magnitude by afactor of 1:3  2:9, i.e., it produces a greater amount ofcancellation between Vee and Ven. The inclusion of cor-relation does not change the HF interpretation of Hund’s rule.It is summarized as follows: The stabilization of thehighest multiplicity state of all the atoms calculated isascribed to the greatest jVenj that is gained at the cost ofincreasing Vee to its highest possible extent. Exchange in thehighest multiplicity state reduces most effectively the Hartreescreening of the nuclear charge at short interelectronicdistances. The resulting contraction of the electron densitydistribution around the nucleus gives rise to the greatest jVenj.Hund’s rule is not ascribed to the exchange energy gain, butto exchange-induced less screening.By means of CASSCF we has also ascertained that thesame conclusion holds for CH2;28) E ¼ 0:0618, T ¼0:0618, V ¼ 0:1235, Ven ¼ 0:1230, Vee ¼ 0:0161,Vnn ¼ 0:0166, andV=T ¼ 2:000; all energies are inhartree units. It is to be noted that spin-polarization in theground state of CH2 lowers Vnn through concomitant changesin the molecular geometries, i.e., the equilibrium internuclearseparations and the equilibrium bond angles. Variationaloptimization of the molecular geometries can be allowed forthrough the fulfillment of the virial theorem in the presentCASSCF.3. Chemical Bonding in H2A list of the binding energy, its components, the virialratio, and the equilibrium internuclear separation evaluatedfrom various methods is given for H2 in Table 1. Both thebinding energy and the equilibrium internuclear separation inDMC40) are in good agreement with those by Kolos andWolniewicz41) or experimental values.42,43) HF underesti-mates the binding energy by about 23% and the equilibriuminternuclear separation by 1% for lack of correlation.40) Thevirial ratio V=T is accurate to the significant figure 2.0for DMC and 2.00 for HF.40)An increase in T ,T (T > 0) comes from an increase inthe curvature of the two-electron wavefunction induced by aspatial variation of the combined attractive Coulomb field ofthe two nuclei separated by Req and by the correlated motionof two electrons. A lowering in V , V (V < 0) is ascribedto a lowering in Ven (Ven < 0) that overwhelms an increasein Vee and Vnn (Vee > 0 and Vnn > 0).As can be seen from Table 1, violation of the virialtheorem in HL, LCAO, and minimal basis full CI gives thewrong sign of T and V .The application of the so-called scaling method enablesone to give each improved version of HL, LCAO, andminimal basis full CI that obeys the virial theorem. A list ofvarious quantities for these improved versions with theiroptimized scaling factors is given at the bottom of Table 1.The scaling method above changes both the sign of T andthat of V involving a reduction in Req.A comparison of the electron density distribution nðrÞ ismade between HL, LCAO, minimal basis full CI, HF, andDMC along the molecular axis z of H2 in Fig. 2 (a), (b), (c),and (d). HF gives a good description of nðrÞ as a whole. Thedifference in nðrÞ between HF and DMC is rather small.Each of HL, LCAO, and minimal basis full CI under-estimates nðrÞ around the two nuclei and overestimates nðrÞin the remote region far away from the two nuclei. This isbecause any approximation using 1s atomic orbitals alone isunable to give an appropriate description of such an essentialreconstruction of nðrÞ as is caused by the combined attractiveCoulomb field of the two nuclei separated by Req. The scalingmethod could give a description of nðrÞ resembling the HFvalue.Correlation in H2 gives a couple of significant changes innðrÞ involving an increase in the equilibrium internuclearseparation Req.40) Firstly, correlation enhances nðrÞ in the leftand right anti-binding regions;44) the value of nðrÞ at theposition of each nucleus is 0.466 bohr3 for DMC and0.460 bohr3 for HF. Secondly, correlation reduces nðrÞ inthe binding region44) intervening between the nuclei; thevalue of nðrÞ at the middle point between the nuclei isFig. 1 Energy differences between high-S and low-S states for the 2p and3p atoms evaluated from HF and DMC.Correct Interpretation of Hund’s Rule and Chemical Bonding Based on the Virial Theorem 6630.284 bohr3 for DMC and 0.288 bohr3 for HF. Corre-sponding to these changes in nðrÞ, the equilibrium internu-clear separation is increased by an amount of about 1%, i.e.,RDMCeq ¼ 1:401 bohr and RHFeq ¼ 1:386 bohr.These correlation-induced features can be interpreted asfollows: The presence of the Coulomb hole around eachelectron reduces the Hartree screening of the nuclear chargeat short interelectronic distances. This may be termedcorrelation-induced less screening. Owing to less screening,nðrÞ is enhanced in the left and right anti-binding regions. Onthe other hand, the same less screening in the binding regionis overwhelmed by an increase in Req to give a reduction ofnðrÞ therein. A set of interconnected correlation effects aboveare reasonable in the light of the electrostatic theorem45)which is closely related to the virial theorem.4. Concluding RemarksThe virial theorem and the Pauli exclusion principle are themost fundamental requirements for the study of many-electron systems. In accordance with the two requirementswe have interpreted Hund’s empirical rule for the six atomsin the second and third rows in the periodic table and for themethylene molecule, taking full account of correlation. Thetraditional interpretation of Hund’s rule allows only for thePauli exclusion principle, completely neglecting the virialtheorem. Both of the two are indispensable for the correctinterpretation. We have analyzed the influence of correlationon chemical bonding in the hydrogen molecule, withemphasis on the fact that the truth of chemical bonding doesnot lie in between the two naive models that resort to 1satomic orbitals alone.AcknowledgmentsThe computation in this work has been performed usingthe facilities of the Center for Computational MaterialsScience (IMR; Institute for Materials Research, TohokuTable 1 The binding energy, its components, the virial ratio, and the equilibrium internuclear separation evaluated from various methods.The exact values by Kolos and Wolniewicz41Þ and experimental values42;43Þ are also given. Energies are measured in units of hartree. Theequilibrium internuclear separation is measured in units of bohr. The figure in each parenthesis indicates the statistical error in the lastdigit.E T V Ven Vee Vnn V=T ReqHL 0:11597 0:1623 0.0463 1:0611 0.4986 0.6088 0.285 1.643LCAO 0:09908 0:2033 0.1042 1:0723 0.5527 0.6238 0.513 1.603minimal basis full CI 0:11865 0:1838 0.0652 1:0435 0.5092 0.5995 0.355 1.668HFa) 0:13365ð2Þ 0.1333(1) 0:2669ð1Þ 1:6494ð1Þ 0.66113(2) 0.7214 2.003(1) 1.386DMCb) 0:17447ð4Þ 0.1749(14) 0:3518ð19Þ 1:6535ð17Þ 0.5880(4) 0.7137 2.011(19) 1.401Exactc) 0:17447 0.1745 0:3489 2.000 1.401Experimentd) 0:17447 1.401HL( ¼ 1:193) 0:13908 0.1391 0:2782 1:5659 0.5805 0.7072 2.000 1.414LCAO( ¼ 1:166) 0:12823 0.1282 0:2565 1:6338 0.6555 0.7218 2.000 1.385minimal basis full CI( ¼ 1:194) 0:14794 0.1479 0:2959 1:6070 0.6120 0.6991 2.000 1.430a) Ref. 40)b) Ref. 40)c) Ref. 41)d) Ref. 42, 43)Fig. 2 The electron density distribution nðrÞ evaluated from (a) HL, (b) LCAO, minimal basis full CI, (c) HF, and (d) DMC along themolecular axis z of H2.664 K. Hongo, T. Oyamada, Y. Maruyama, Y. 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Correct Interpretation of Hund's Rule and Chemical Bonding Based on the Virial Theorem

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Correct Interpretation of Hund\u27s Rule and Chemical Bonding Based on the Virial Theorem

Correct Interpretation of Hund\u27s Rule and Chemical Bonding Based on the Virial Theorem

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