The electronic stress tensor of the hydrogen molecule ion H_2^+ is
investigated for the ground state (sigma_g 1s) and the first excited state
(sigma_u^* 1s) using their exact wave functions. A map of its largest
eigenvalue and corresponding eigenvector is shown to be closely related to the
nature of chemical bonding. For the ground state, we also show the spatial
distribution of interaction energy density to describe in which part of the
molecule stabilization and destabilization take place.Comment: 9 pages, 4 figure

Akitomo Tachibana

Kazuhide Ichikawa

W. Pauli

English

arXiv

Kazuhide  Ichikawa

Crossref

Stress tensor of the hydrogen molecular ion

Ichikawa, Kazuhide

Tachibana, Akitomo

Kyoto University Research Information Repository

Physical Review A

The electronic stress tensor of the hydrogen molecule ion H+2 is investigated for the ground state (σg1s) and the first excited state (σu∗1s) using their exact wave functions. A map of its largest eigenvalue and corresponding eigenvector is shown to be closely related to the nature of chemical bonding. For the ground state, we also show the spatial distribution of interaction energy density to describe in which part of the molecule stabilization and destabilization take place

Title Stress tensor of the hydrogen molecular ionAuthor(s)Ichikawa, Kazuhide; Tachibana, AkitomoCitationPHYSICAL REVIEW A (2009), 80(6)Issue Date2009-12URL http://hdl.handle.net/2433/109847Right© 2009 The American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityStress tensor of the hydrogen molecular ionKazuhide Ichikawa and Akitomo Tachibana*Department of Micro Engineering, Kyoto University, Kyoto 606-8501, JapanReceived 19 August 2009; published 7 December 2009The electronic stress tensor of the hydrogen molecule ion H+2 is investigated for the ground state g1s andthe first excited state u1s using their exact wave functions. A map of its largest eigenvalue and correspond-ing eigenvector is shown to be closely related to the nature of chemical bonding. For the ground state, we alsoshow the spatial distribution of interaction energy density to describe in which part of the molecule stabiliza-tion and destabilization take place.DOI: 10.1103/PhysRevA.80.062507 PACS numbers: 31.10.z, 31.15.ae, 03.65.wThe stress tensors are used widely for description of in-ternal forces of matter. They have been also established inquantum mechanical context 1, although it seems that basicidea dates back to Ref. 2. Later, applications for quantumsystems are investigated in many literatures from various as-pects 3–22. For example, Ref. 6 and followers have fo-cused on stress tensor, which is associated with forces onnucleus. Our focus in this paper is electronic stress tensors,which measure effects caused by internal forces acting onelectrons in molecules and particularly those betweenbonded atoms 12,15,16. Such electronic stress tensor hasbeen successfully used to define bond orders expressing thestrength of chemical bonds 18. It should be noted thatpeople sometimes use different definitions for the stress ten-sor due to their own purpose and formalism. We use theexpression derived by one of the authors in Ref. 12, whichwe believe most straightforward and suitable to discusschemical bonds.In this paper, we elucidate the role of the electronic stresstensor using hydrogen molecule ion H+2, the simplest of allmolecules, whose exact wave functions are known. Althoughthere have been analyses of electronic stress tensor of othermolecules using approximate wave functions expanded inthe Gaussian functions 12,15,16,18,19,22, analysis usingexact wave functions has not been conducted. We calculatespatial distributions of stress tensor and other associatedquantities from the exact wave functions of H+2 and examinetheir detailed features that can be used to illustrate the chemi-cal bond of the molecule. We believe that this paper demon-strates usefulness of the electronic stress tensor for an analy-sis of chemical bonding in the simplest but most exact form.1We begin by showing our expression of the electronicstress tensor ijS constructed from the H+2 wave function .That is,ijS=24mi j − i j + c.c. , 1where i , j= 1,2 ,3 denote spatial coordinates, m is theelectron mass, and c.c. stands for complex conjugate. Theexpression for many-electron systems, that is, for almost ev-ery other molecule, is found in Ref. 12. There, the stresstensor has been derived by field theoretic method which isapplicable to many-particle system. Here, for illustrative pur-pose, we derive it using the Schrödinger equation for a singleelectron in H+2.The time-dependent Schrödinger equation readsit= −22m2 + V , 2where the potential energy is assumed to be real. It is well-known that the continuity equation n /t+ · j=0 holds bydefining probability density n= 2 and probability fluxj=  /mIm. Now, the equilibrium equation for elec-tronic stress tensor is obtained from an equation of motion ofj. Namely, using Eq. 2, time derivative of j can be writtenasjt= −24m2 · 2 −  2 + c.c. −2m V .For the steady state, which is the case of molecular systemswe consider, since j /t=0, we obtain the equilibrium equa-tion jijS + FL,i = 0, 3where FL,i=−2iV is the Lorentz force generated by nu-clei. For later convenience, we here define the first term inthe equation as tension F,i jijS. From this equation, wesee that ijS is the stress caused by purely quantum mechani-cal effect.We compute the stress tensor defined by Eq. 1 using theexact wave functions of H+2 25. As is usually done for thestress tensor, we examine its largest eigenvalue and corre-sponding eigenvector. The sign of the largest eigenvalue tellswhether electrons at a certain point in space feel tensile forcepositive eigenvalue or compressive force negative eigen-value and the eigenvector tells direction of the force.The case for the ground state g1s is plotted in the planeincluding two H nuclei as shown in the left panel of Fig. 1.The origin of the coordinate is taken to be the midpoint ofthe two nuclei. The eigenvalue along the internuclear axis isalso plotted as red solid line in Fig. 2. The internuclear dis-tance is taken to be 2.0 bohrs, which gives the equilibriumdistance Re to be more precise, Re=1.9972 bohrs 26 but it*akitomo@scl.kyoto-u.ac.jp1Calculations of stress tensor field as regards a H+2 molecule arefound in Refs 23,24. but with different definitions from ours.PHYSICAL REVIEW A 80, 062507 20091050-2947/2009/806/0625074 ©2009 The American Physical Society062507-1does not make practical difference in our argument below.The region with positive eigenvalue spreads between the nu-clei, which corresponds to the formation of a chemical bond.The electronic stress tensor has this useful feature that it canexpress something is pulled up that is, tensile force existswhen a covalent bond is formed. In detail, the positive ei-genvalue region is bounded by a closed spherelike surfacethat touches two H nuclei. Also, in that region, the eigenvec-tors forms a bundle of flow lines that connects the H nuclei.Such a region, called “spindle structure” 15, is clearly vis-ible using the analysis with the exact wave function.The case for the first excited state u1s is shown in theright panel of Fig. 1 on the same plane and with the sameinternuclear distance. See also the green dashed line in Fig. 2for the eigenvalue along the internuclear axis. Since this statedoes not form a bound state, we do not expect a bond be-tween the H nuclei. Indeed, there is no region with positiveeigenvalue between the H nuclei for this case. In addition,eigenvectors are perpendicular to the internuclear axis at theregion around the middle of the two H nuclei, showing thereis no force connecting them.These results, positive eigenvalue for bonding orbital andnegative eigenvalue for antibonding orbital, are consistentwith Ref. 15, where these features are pointed out using H2and He2, respectively.We now examine the tension field F,i jijS of the H+2molecule. Writing explicitly,F = −24m · 2 −  2 + c.c. . 4Since this force, which cancels the classical Lorentz force ateach point in space Eq. 3, also expresses purely quantummechanical effects, it is considered to carry some informa-tion of a chemical bond. The spatial distribution of tensionvector is plotted for g1s and u1s states in Fig. 3. On com-paring Figs. 1 and 3, we see that in contrast with the case ofthe eigenvalue and eigenvector of the stress tensor, there isnot much difference between the g1s state and the u1sstate for the case of the tension vector. In Fig. 1, there is astriking difference between g1s and u1s that the formerdevelops a region with positive eigenvalue while the latterexhibit negative region for the most of the space. There isalso a marked difference in the pattern of eigenvectors at theinternuclear region. In Fig. 3, the two states show a relativelysimilar pattern of the tension vector field. One difference canbe pointed out that there is a point x ,z= 0.0,0.0 at whichtension vanishes for the g1s state while the u1s states hasa plane z=0 on which it vanishes. However, overall featureresembles closely to each other. This indicates that the stresstensor has much more information as regards a chemicalbond.We note, however, that the tension can provide a handyway to characterize a chemical bond. Actually, it has beenused to define an index of bond strength and turned out to be-0.0100.010.020.030.040.050.06z (bohrs)x(bohrs)-2 -1 0 1 2-2-1012-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.0100.01z (bohrs)x(bohrs)-2 -1 0 1 2-2-1012(b)(a)FIG. 1. Color online The spatial distributions of the largest eigenvalue and corresponding eigenvectors of stress tensor of H+2 moleculeare plotted in the plane including two H nuclei. They are located at z ,x= −1.0,0.0 and 1.0, 0.0. The left panel is for the ground stateg1s and the right panel is for the first excited state u1s. The black solid line shows the zero surface of the eigenvalue. For g1s state,it is positive inside the circle and is negative outside. For the u1s state, it is negative on both sides of the surface.-0.1-0.08-0.06-0.04-0.0200.020.040.060.08-3 -2 -1 0 1 2 3Thelargesteigenvaluez (bohrs)FIG. 2. Color online The largest eigenvalue of the stress tensoralong the internuclear axis along the z axis in Fig. 1. The red solidline is for the ground state g1s and the green dashed line is forthe first excited state u1s.KAZUHIDE ICHIKAWA AND AKITOMO TACHIBANA PHYSICAL REVIEW A 80, 062507 2009062507-2very useful 18,19,22. In Ref. 18, it has been proposed tocalculate energy density defined below, Eq. 6 at the“Lagrange point,” which is defined as the vanishing point ofthe tension such as we have found at the midpoint of theinternuclear axis of the ground state H+2 molecule in Fig. 3.It should also be noted that the existence of the Lagrangepoint alone cannot guarantee a chemical bond as is easilyseen from the tension of the u1s state of H+2, which has theLagrange point or rather the Lagrange plane z=0. Hence,the tension can be a useful tool when combined with analysisof the stress tensor.Finally, we describe our definition of energy density,which is derived from the stress tensor, and discuss the in-teraction energy density of the ground state of the H+2 mol-ecule. In Ref. 12, it has been proposed to define energydensity  from the trace of the stress tensor as12	i iiS 5=28m2 −  ·  + c.c. . 6Note that this definition gives correct total energy when in-tegrated over the whole space and the virial theorem is ap-plied. Furthermore, we consider the difference of the energydensity between the ground state and the separated atoms.We call this interaction energy density  18, and it is auseful quantity to discuss where in a molecule, stabilizationor destabilization take place by chemical bond formation.For the case of H+2, it is appropriate to define  as = H+2 −HA + HB2, 7where HA denotes the H atom at z=−1.0 and HB at z=1.0 asnoted above, more precise Re is 0.3% shorter and the virialtheorem holds only at the equilibrium position but we canpractically consider Re to be 2.0 bohrs. We plot this in theleft panel of Fig. 4. From the definition, negative  shownin blue inside the solid line indicates stabilized regions andpositive  shown in red outside the solid line indicatesdestabilized regions. We see that stabilization mostly takesplace between the H nuclei. The stabilized region slightlyexpands outside of the internuclear region and the zero sur-face of  is found to include the two nuclei.It is instructive to compare  with the differential elec-tron density n,n = nH+2 −nHA + nHB2, 8where n= 2 is the ordinary electron density. This is plottedin the right panel of Fig. 4. We see that there is a region withincreased electron density shown in red between the H nu-clei and one with decreased electron density shown in blueoutside the internuclear region. Compared with the map of, the region with increased electron density correspondsnicely to the stabilized region, which is consistent with aconventional picture of chemical bond formation. However,it should also be noticed that  and n do not have one-to-one correspondence. For example, we see in Fig. 4 that has slightly larger zero surface that that of n. More-over, integrating n over the whole space gives zero whereas gives the difference of binding energy between H+2 andH atom.In conclusion, we have calculated the stress tensor of thesimplest molecule, H+2, and demonstrated its relevance tochemical bonding. It is not only the simplest but is also mostaccurate due to the existence of the exact wave functions.By using them, we are able to investigate the eigenvalueand eigenvector of the stress tensor and the tension vectorfield of both the ground state and the first excited state in avery accurate manner. It also makes possible the precisecomparison between the interaction energy density, which012345z (bohrs)x(bohrs)-2 -1 0 1 2-2-1012012345z (bohrs)x(bohrs)-2 -1 0 1 2-2-1012(b)(a)FIG. 3. Color online The spatial distributions of the tension F of H+2 molecule are plotted similarly to Fig. 1. The left panel is for theground state g1s and the right panel is for the first excited state u1s. Normalized tension vectors F / F are shown by arrows and thenorm F is depicted by a color map. F diverges at the positions of the nuclei. Contours which correspond to F=5 are drawn by blackdashed lines. F=0 at 0.0, 0.0 for the g1s state and on z=0 for the u1s state, where arrows are not shown.STRESS TENSOR OF THE HYDROGEN MOLECULAR ION PHYSICAL REVIEW A 80, 062507 2009062507-3tells the stabilized/destabilized regions, and conventional dif-ferential electron density with respect to the ground state. Webelieve that this calculation can be very useful as the mostbasic example which shows an important role of the elec-tronic stress tensor for illustrating a chemical bond. More-over, its exactness would serve as a benchmark to test valid-ity of the calculation of the stress tensor which usesapproximate wave functions. This is also very importantsince we can only obtain approximate wave functions foralmost every molecular system. Therefore, future investiga-tion should include how the stress tensor of H+2 deviatesfrom the exact form as we change the level of approximationof the quantum chemistry calculation to derive H+2 wavefunctions. Such a study will solidify the basis of the stresstensor analysis of molecular systems and open up areas offurther application.1 W. Pauli, General Principles of Quantum MechanicsSpringer-Verlag, New York, 1980.2 E. Schrödinger, Ann. Phys. 386, 109 1926.3 S. T. Epstein, J. Chem. Phys. 63, 3573 1975.4 R. F. W. Bader, J. Chem. Phys. 73, 2871 1980.5 A. S. Bamzai and B. M. Deb, Rev. Mod. Phys. 53, 95 1981.6 O. H. Nielsen and R. M. Martin, Phys. Rev. Lett. 50, 6971983.7 O. H. Nielsen and R. M. Martin, Phys. Rev. B 32, 37801985.8 N. O. Folland, Phys. Rev. B 34, 8296 1986.9 N. O. Folland, Phys. Rev. B 34, 8305 1986.10 M. J. Godfrey, Phys. Rev. B 37, 10176 1988.11 A. Filippetti and V. Fiorentini, Phys. Rev. B 61, 8433 2000.12 A. Tachibana, J. Chem. Phys. 115, 3497 2001.13 A. Martín Pendás, J. Chem. Phys. 117, 965 2002.14 C. L. Rogers and A. M. Rappe, Phys. Rev. B 65, 2241172002.15 A. Tachibana, Int. J. Quantum Chem. 100, 981 2004.16 A. Tachibana, J. Mol. Model. 11, 301 2005.17 S. Morante, G. C. Rossi, and M. Testa, J. Chem. Phys. 125,034101 2006.18 P. Szarek and A. Tachibana, J. Mol. Model. 13, 651 2007.19 P. Szarek, Y. Sueda, and A. Tachibana, J. Chem. Phys. 129,094102 2008.20 J. Tao, G. Vignale, and I. V. Tokatly, Phys. Rev. Lett. 100,206405 2008.21 P. W. Ayers and S. Jenkins, J. Chem. Phys. 130, 1541042009.22 P. Szarek, K. Urakami, C. Zhou, H. Cheng, and A. Tachibana,J. Chem. Phys. 130, 084111 2009.23 A. S. Bamzai and B. M. Deb, Int. J. Quantum Chem. 20, 13151981.24 M. J. Godfrey, J. Phys. B: At. Mol. Opt. Phys. 23, 24271990.25 D. R. Bates, K. Ledsham, and A. L. Stewart, Philos. Trans. R.Soc. London, Ser. A 246, 215 1953.26 L. J. Schaad and W. V. Hicks, J. Chem. Phys. 53, 851 1970.-0.1-0.08-0.06-0.04-0.020z (bohrs)x(bohrs)-2 -1 0 1 2-2-1012-0.0100.010.020.030.040.050.06z (bohrs)x(bohrs)-2 -1 0 1 2-2-1012(b)(a)FIG. 4. Color online The spatial distributions of the interaction energy density  left panel and the differential electron density nright panel of H+2 molecule are plotted for the ground state g1s. The zero surface is shown by the red solid line for each panel. Theinteraction energy diverges at the positions of the nuclei and contours that correspond to =−0.1 are shown by white dashed lines in the leftpanel.KAZUHIDE ICHIKAWA AND AKITOMO TACHIBANA PHYSICAL REVIEW A 80, 062507 2009062507-4

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Stress Tensor of the Hydrogen Molecular Ion

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