Electron Chirality in Amino Acid Molecules
aa r X i v : . [ phy s i c s . a t m - c l u s ] A p r Electron Chirality in Amino Acid Molecules
Masato Senami ∗ and Tomoki Shimizu Department of Micro Engineering, Kyoto University, Kyoto 615-8540, Japan (Dated: April 17, 2020)We evaluated the total electron chirality in alanine, serine, and valine, which are molecules thathave chiral structures. Previously, it has been considered that the total electron chirality of moleculescomposed of only light elements cannot be computed within usual computational conditions ofrelativistic four-component wave functions. In this work, it is shown that the total electron chiralitycan be calculated if some diffuse functions are added to Gaussian basis sets. This is demonstrated forthe H O molecule. By adding diffuse Gaussian functions to basis sets, the total electron chiralityof L-alanine, L-serine, and L-valine are evaluated. It is also shown that the total electron chiralityis derived by the cancellation between large contributions from each orbital, and the total electronchirality in excited and ionized states is expected to be much larger than that of the ground state. PACS numbers: 31.30.J-, 31.30.jg, 95.30.Ft
I. INTRODUCTION
Electrons are chiral objects from the viewpoints of thefundamental physics. The weak interaction, which is oneof the four fundamental interactions in particle physics,produces larger reaction rates for left-handed particlesthan for right-handed particles [1]. Electron is a pointparticle, and it is considered to be spherically symmetric.However, electron has velocity and spin angular momen-tum, and the chirality of the electron can be defined onthe basis of the relativistic form of these quantities.In our previous works [2, 3], it has been establishedthat electron chirality is biased toward a particularform, L- or D-configuration, in chiral molecules ow-ing to spin-orbit interactions and the electron chiralityin L-configuration molecules is opposite of that in D-configuration molecules. As mentioned above, the in-teraction of electrons is dependent on electron chirality.Hence, L- and D-enantiomers exhibit different interactionproperties in the weak interaction.Electron chirality in chiral molecules (ECCM) may berelated to some unsolved problems. One example is thechiral-induced spin selectivity (CISS) effect (e.g., [4]),which is the polarization of the electron spin through chi-ral organic molecules. It is unclear how this polarizationis generated through chiral molecules. For example, someresearchers consider that spin-orbit interactions inducethis effect; however a two orders of magnitude larger spin-orbit interaction than that of a carbon atom is needed.It has been shown that electron chirality is one of the ori-gins of the torque for the electron spin in the quantumfield theoretical picture [5, 6]. Therefore, ECCM may bea good solution for CISS.Another example is the homochirality in nature. In ourprevious paper [2], we pointed out that ECCM may bethe solution of homochirality. On Earth, amino acids andsugars are extremely biased in chirality. Almost all amino ∗ Electronic address: [email protected] acids have L-configuration, while sugars predominantlyexist in D-configuration. It is believed that a small sourceof enantiomeric asymmetry is produced; then, this enan-tiomeric excess is enhanced to large imbalance throughchiral amplification processes. It is unclear how this ini-tial bias is generated during the evolution of the universeor Earth. Thus, far several scenarios for the productionof this initial imbalance have been proposed [7]. For ex-ample, this imbalance has been proposed to be producedby circularly polarized light from pulsars, the parity-violating energy difference between enantiomers by theweak interaction, or left-handed electrons by beta decayof nuclei (the Vester-Ulbricht hypothesis [8]). In our pre-vious paper, we added ECCM to this list [2]. One formof an enantiomeric pair has different reaction rates of theweak interaction owing to ECCM because left-handedelectrons interact more strongly than right-handed onesby the weak interaction. Thus, one-configuration of anenantiomeric pair is more fragile than the rest of the pairby the interactions with particles such as cosmic rays andneutrinos produced by nuclear fusion in stars. This givesinevitably nonzero contribution to the difference in thenumber of enantiomers.These two examples are driven by molecules consist-ing of only light elements up to the second row in theperiodic table. In our previous work [2], the reliable re-sult of the total chirality of H O , which is one of thesimplest examples of chiral molecules consisting of onlylight elements, could not be obtained by ordinary four-component relativistic quantum structure computations.To study these two examples, a reliable computationalmethod is mandatory.In this study, first, we establish a computationalmethod of the total chirality of chiral molecules consist-ing of only light elements. For this purpose, an H O molecule is used, and computations are determined to bereliable if the dependence of the total chirality on the di-hedral angle is the same as that of other H X molecules.Because the parity-violating energy difference, which pri-marily depends on the electron chirality at the positionsof nuclei, was accurately calculated [3], computed wavefunctions around nuclei are sufficiently accurate. Thus,the addition of diffuse functions to basis sets is specu-lated to be effective. In this paper, we will confirm thisspeculation. Second, the total electron chirality of aminoacid molecules is calculated for alanine, serine, and va-line using basis sets containing additional diffuse func-tions. For our scenario of the generation of homochiral-ity, L-enantiomers of amino acid molecules should have asmaller ratio of left-handed electrons, which correspondsto the positive value of chirality.This paper is organized as follows. In the next sec-tion, the definition of the total electron chirality and ourcomputational method are explained. In Sec. III, our re-sults are presented. The dependence of the total electronchirality of H O on the dihedral angle is calculated forvarious basis sets, and it is established that the additionof diffuse functions is effective for the computation of thetotal electron chirality of chiral molecules consisting ofonly light elements. Then, the total electron chirality ofL-alanine, L-serine, and L-valine is evaluated; it is de-termined that all of the above-mentioned molecules, andthey all exhibit positive chirality values. The total elec-tron chirality is shown to be the result of the cancellationbetween large contributions from each orbital. The lastsection is devoted to our summary. II. THEORY AND COMPUTATIONAL DETAILS
Electron chirality is classified into right- or left-handedelectrons. The right- and left-handed projection opera-tors of the electron chirality are defined by: P R = 1 + γ , P L = 1 − γ , (1)where γ = iγ γ γ γ and γ µ ( µ = 0 −
3) are the gammamatrices. In another expression, γ = 13 X i =1 α i Σ i (2)where α i = γ γ i and Σ i is the 4 × γ , exhibitsthe characteristic of the parity odd [1, 9]. The right-and left-handed electrons are described by the followingprojection operators: ψ R ( ~r ) = P R ψ ( ~r ) , ψ L ( ~r ) = P L ψ ( ~r ) , (3)where ψ is the four-component wave function of the elec-tron. The difference in the density between the right-handed and left-handed electrons is described by the chi-rality density as follows: ψ † ( ~r ) γ ψ ( ~r ) = ψ † R ( ~r ) ψ R ( ~r ) − ψ † L ( ~r ) ψ L ( ~r ) . (4)In the computational chemistry field, chirality densitywas first taken into account in Ref. [10] after many text-books on relativistic quantum theory explained electron chirality. The total electron chirality is obtained by inte-grating the chirality density over an entire molecule, Z d ~rψ † ( ~r ) γ ψ ( ~r ) . (5)The total electron chirality is a different notion from theparity-violating energy difference (e.g., see [11]). Theparity-violating energy difference depends only on theelectron chirality at the positions of nuclei in a molecule,while the total electron chirality is obtained by integrat-ing the entire molecule.Electronic structure should be computed in a fully rela-tivistic form, i.e., a four-component wave function. Thiscomputation was carried out using the DIRAC17 pro-gram package [12]. We adopted the relativistic Hartree-Fock method with the Dirac-Coulomb Hamiltonian. Forthe derived wave functions, the total electron chiralitywas calculated by the QEDynamics program package[13]. For basis sets, we chose dyall.ae3z, dyall.ae4z, cc-pV5Z, and cc-pV6Z [14–16]. For all basis sets, uncon-tracted forms were used. The number of Gaussian func-tions of cc-pV5Z (cc-pV6Z) in the uncontracted form isapproximately similar to that of dyall.ae3z (dyall.ae4z).Correlation consistent basis sets have more functions forhigher angular momentum functions, such as g or h.However, correlation consistent basis sets are not con-structed with relativistic effects.Moreover, diffuse functions are added to these basissets. For Dyall basis sets (dyall.ae3z and dyall.ae4z),nomenclature, such as q-aug-dyall.ae3z, is adopted, anddiffuse functions are added as follows. For each angularmomentum, the exponent factor ratio is determined bythe ratio of the exponents between the two most diffuseexponent basis functions. If only one function exists inthe angular momentum, the ratio is chosen to be 1/3.5.Using this ratio, new exponent functions are repeatedlyadded n times, where n = 2, 3, 4 for X =d, t, q in X -aug-dyall.ae m z basis sets ( m = 3, 4). If atomic species arespecified as a prefix of a basis set name ( Atom )- X -aug-dyall.ae m z, diffuse functions are added only to the basisfunctions of specified atoms. For correlation-consistentbasis sets, ordinary augmented basis sets are adopted,i.e., aug-cc-pV5Z, and aug-cc-pV6Z [16, 17], which addone small exponent function to each angular momentum.In our computations, H O , H Te , alanine, serine,and valine were calculated. The geometrical structureof H X (X=O, Te) was set to be the same as that inour previous work [2]. For alanine, serine, and valinemolecules, the geometrical optimization was performedwith the dyall.ae2z basis set [14]. The structure of themolecules is shown in Fig. 1. The dihedral angle φ ofH X is defined in this figure. For the study of varying φ , only φ is varied from this optimized geometry, whilethe other parameters, such as internuclear lengths, arefixed.In the following, we adopted the atomic unit unlessstated otherwise. (a) H X (b) Alanine(c) Serine(d) Valine FIG. 1: Structures of (a)H X , (b) alanine, (c) serine, and(d) valine molecules and the definition of the dihedral angle φ of H X . -2.5 × 10 −8 -2.0 × 10 −8 -1.5 × 10 −8 -1.0 × 10 −8 -5.0 × 10 −9 × 10 × 10 −9 × 10 −8 × 10 −8 × 10 −8 × 10 −8
0 60 120 180 240 300 360 i n t eg r a t ed c h i r a li t y den s i t y dihedral angle [degrees] dyall.ae3zdyall.ae4zcc-pV5Zcc-pV6Z H O -2.0 × 10 −5 -1.5 × 10 −5 -1.0 × 10 −5 -5.0 × 10 −6 × 10 × 10 −6 × 10 −5 × 10 −5 × 10 −5
0 60 120 180 240 300 360 i n t eg r a t ed c h i r a li t y den s i t y dihedral angle [degrees] dyall.ae4z H Te FIG. 2: Total chirality of H O and H Te as a function ofthe dihedral angle, φ . III. RESULTSA. H O Figure 2 shows the total electron chirality of H O andH Te for the basis sets without additional diffuse func-tions. The dependence of the total chirality of H O onthe dihedral angle, φ , should exhibit the same curve asthat of H Te . This occurs because the molecule struc-ture governs the total chirality, as reported in our pre-vious works [2, 3], where H Se , H S , and H Te ex-hibited the same dependence on φ . However, the resultfor H O shows a different dependence on φ from thatof H Te . Moreover, the curves of H O differ from eachother. This indicates that this result does not have areliable accuracy.For the total electron chirality, the accuracy was notsufficient; however, these computations were sufficientlyaccurate to represent the parity-violating energy differ-ence. In our previous study [2, 3], the parity-violating en-ergy difference of H O exhibited the same dependence asthat of other H X molecules on φ . The parity-violatingenergy difference of H X molecules is predominantly de-termined by the electron chirality density at the X nu-cleus position. Hence, our wave functions are sufficientlyaccurate around the nuclei, and the accuracy is not suf-ficient in regions away from nuclei. To confirm that the -5.0 × 10 −6 -4.0 × 10 −6 -3.0 × 10 −6 -2.0 × 10 −6 -1.0 × 10 −6 × 10 × 10 −6 × 10 −6 × 10 −6 × 10 −6 × 10 −6
0 60 120 180 240 300 360-3.0 × 10 −2 -2.0 × 10 −2 -1.0 × 10 −2 × 10 × 10 −2 × 10 −2 × 10 −2 M PV O [ a . u .] M PV T e [ a . u .] dihedral angle [degrees] dyall.ae3zdyall.ae4zcc-pV5Zcc-pV6Zdyall.ae4z (Te) FIG. 3: Electron chirality densities at the positions of O andTe nuclei in H O and H Te as a function of φ . -4.0 × 10 −9 -3.0 × 10 −9 -2.0 × 10 −9 -1.0 × 10 −9 × 10 × 10 −9 × 10 −9 × 10 −9 × 10 −9
0 60 120 180 240 300 360 i n t eg r a t ed c h i r a li t y den s i t y dihedral angle [degrees] q-aug-dyall.ae3zq-aug-dyall.ae4zaug-cc-pV5Zaug-cc-pV6ZO+q-aug-dyall.ae4zO+t-aug-dyall.ae4zO+d-aug-dyall.ae4z FIG. 4: Total electron chirality of H O as a function of thedihedral angle, φ , with additional diffuse functions. wave functions in this study were accurate around O nu-clei, the electron chirality densities at O and Te nuclei, M XPV , are shown in Fig. 3. For all basis sets, the curvesat the O nucleus are consistent with that at the Te nu-cleus. Therefore, our wave functions are confirmed to besufficiently accurate around the nuclei. It is speculatedthat wave functions should be improved in regions awayfrom nuclei. Thus, we add additional diffuse functions tobasis sets.Figure 4 shows the total electron chirality of H O asa function of the dihedral angle, for the basis sets withadditional diffuse functions. It is observed that all resultsshow the same dependence on φ . Specifically, the resultsof dyall.ae4z with additional diffuse functions are well-converged, and sufficient accuracy is attained. The num-ber of additional diffuse functions for Dyall basis sets ismuch larger than that for correlation-consistent ones. Asobserved from the results of correlation-consistent ones,many additional diffuse functions are not required, i.e.,only ordinary augmentation or twice of it is sufficient toevaluate the total electron chirality. The results of cc-pV5Z and cc-pV6Z are very similar to those of dyall.ae3zand dyall.ae4z, respectively. Hence, the relativistic ef-fect in basis sets is seen to be less important than otherrelativistic effects. The spin-orbit interaction of oxygenatoms is very small compared to S, Se, and Te atoms, TABLE I: Contribution of each orbital to the total electronchirality of H O at φ = 45 ◦ for the result of q-aug-dyall.ae3z. E i is the orbital energy and ψ † i ( ~r O ) γ ψ i ( ~r O ) is the electronchirality at the position of an oxygen nucleus. The value ofthe orbital contribution to the electron chirality is shown forone electron in Kramers pair, while both values in pairs aresummed for the total.No. E i [a.u.] ψ † i ( ~r O ) γ ψ i ( ~r O ) R d xψ † i γ ψ i − . − . × − − . × − − . − . × − − . × − − . − . × − . × − − .
214 1 . × − . × − − . − . × − − . × − − .
674 2 . × − . × − − . − . × − − . × − − .
548 4 . × − − . × − − . − . × − . × − Total − . × − − . × − whose total electron chirality was studied in our previ-ous works [2, 3]. Accordingly, the accurate inclusion ofthe spin-orbit interaction was challenging, and we accu-rately calculated it with additional diffuse functions.In our previous work [2], we have reported that thetotal electron chirality of H Te was derived as the re-sult of the cancellation between large contributions fromeach orbital. Actually, many higher energy orbitals,such as the highest occupied molecular orbital (HOMO),have larger contribution than the total electron chiral-ity. Hence, it was speculated that if chiral moleculeswere excited or ionized, this cancellation was broken andthe chiral molecules had much larger total electron chi-rality. In the work, we confirmed this speculation withimaginary H Te . To confirm that this speculation istrue even for H O , which is a molecule composed ofonly light elements and has small spin-orbit interaction,each orbital contribution to the total electron chirality ofH O is studied. In Table I, the contribution to the totalelectron chirality, R d xψ † i γ ψ i , and that to the electronchirality at the position of an O nucleus, ψ † i ( ~r O ) γ ψ i ( ~r O ),are shown for the result of q-aug-dyall.ae3z of the H O molecule at φ = 45 ◦ , where ψ i is the i -th orbital. Thetotal electron chirality is − . × − . Contributionsfrom each orbital except for the lowest two orbitals areconsiderably larger than even the total value. Therefore,even for the molecules of only light elements, excitationor ionization is considered to enhance the total electronchirality. We will confirm this enhancement with configu-ration interaction computations in our next paper. More-over, this property is also observed for the electron chiral-ity at the position of an O nucleus. The parity-violatingenergy difference, which is dominantly determined by theelectron chirality at the position of an O nucleus, is spec-ulated to be enhanced by excitation or ionization. Thecancellation is not stronger than the total electron chiral-ity, and the enhancement of the parity-violating energydifference is not as large as the total electron chirality.These features are the same as those of H Te . B. Amino Acids
In the previous subsection, we showed that the to-tal electron chirality of light element molecules can becomputed by extending basis sets with additional diffusefunctions. In this subsection, the results for amino acidsare shown.In Table II, the total electron chirality of L-alanine, L-serine, and L-valine is shown for several basis sets. More-over, the parity-violating energy is also shown as a ref-erences. The parity violating energy is calculated by thefollowing equation, E PV = G F √ X n Q nW ψ † ( ~r n ) γ ψ ( ~r n ) , (6)where G F / ( ~ c ) = 1 . × − GeV − [18] and ~r n is theposition of a nucleus, n . Q nW = Z n (1 − θ W ) − N n is the weak charge of a nucleus, n , where Z n and N n are the number of protons and neutrons in the nu-cleus, n , respectively, and θ W is the weak-mixing an-gle, whose value is determined by sin θ W = 0 . O . The wave functions around nuclei are suf-ficiently accurate, while those away from them are notaccurate. Hence, the inclusion of additional diffuse func-tions is required for reliable computations, as shown inthe previous subsection. For the results shown in Ta-ble II, several tens of percent error remain between thetriple and quadruple diffuse functions of alanine and va-line. Within this error, the results are considered to besettled. The values are much smaller than those of H O ,and this difference is considered to originate in compli-cated structures of amino acid molecules. In is notewor-thy that all L-configuration amino acid molecules havepositive electron chirality, where the right-handed elec-tron is the major component. This means that D-aminoacid molecules have a larger rate of the weak interactionbecause the left-handed electron has a larger couplingconstant of the weak interaction than the right-handedone. This characteristic of L-amino acid is consistentwith our scenario.In Table III, the contribution from each orbital to thetotal electron chirality is shown. The total electron chi-rality of all amino acid molecules was derived by the sig-nificant cancellation by two or three orders. This indi-cates that the total electron chirality is strongly enhanced for excited or ionized states. IV. SUMMARY
The total electron chirality in chiral amino acidmolecules has been evaluated. Alanine, serine, and va-line were chosen for these molecules. Previously, the totalelectron chirality of these light element molecules cannotbe calculated owing to the smallness of the spin-orbitinteraction. We speculated that additional diffuse func-tions, which are Gaussian functions with a small expo-nent, are effective for this computation. This speculationwas based on the fact that wave functions around nucleiare sufficiently accurate and those away from them arenot because the results of the parity-violating energy dif-ference are well-converged. In this study, we confirmedour speculation for the H O molecule by numerical com-putation. The results with additional diffuse functionsare well converged and the dependence of the total elec-tron chirality on the dihedral angle is clearly consistentwith our expectation. Thus, we established the computa-tional method of the total electron chirality of moleculesconsisting of only light elements. In this study, the to-tal electron chirality of alanine, serine, and valine wereevaluated with additional diffuse functions. It was de-termined that the values of the total electron chirality ofL-alanine, L-serine, and L-valine were much smaller thanthat of H O . The total electron chirality of all L-aminoacid molecules was positive. This means that the num-ber of right-handed electrons is larger than that of theleft-handed ones in L-amino acid molecules, and D-aminoacid molecules have a larger number of left-handed elec-trons than right-handed ones. Left-handed electron hasa larger coupling constant of the weak interaction and,hence, D-amino acid molecules exhibit a larger reactionrate of the weak interaction than L-configuration. Thus,D-amino acid molecules may be more broken in the evolu-tion of the universe by reactions with cosmic rays or neu-trinos in space. Therefore, ECCM may be the solutionfor the problem of homochirality in nature. In addition,we showed the contribution from each orbital to the totalelectron chirality for these amino acids. We determinedthat the small total electron chirality is derived by thecancellation between large contributions from HOMOs.This indicates that the total electron chirality in excitedand ionized states is much larger than that of the groundstate, as shown for imaginary doubly ionized states in ourprevious work [2]. In our next study, we will confirm theenhancement of the total electron chirality in excited andionized states. For this confirmation, computations withthe configuration interaction method will be performed. Acknowledgments
This work was supported by Grants-in-Aid for Scien-tific Research (17K04982 and 19H05103).
TABLE II: Total electron chirality and the parity violating energy of L-amino acids.molecule basis set R d xψ † γ ψ E PV [a.u.]L-Alanine dyall.ae3z − . × − − . × − ONC+t-dyall.ae3z 1 . × − − . × − ONC+q-dyall.ae3z 2 . × − − . × − dyall.ae4z 6 . × − − . × − L-Serine dyall.ae3z 6 . × − − . × − ONC+t-dyall.ae3z 2 . × − − . × − ONC+q-dyall.ae3z 2 . × − − . × − dyall.ae4z 2 . × − − . × − L-Valine dyall.ae3z − . × − − . × − ONC+t-dyall.ae3z 1 . × − − . × − ONC+q-dyall.ae3z 2 . × − − . × − dyall.ae4z 9 . × − − . × − [1] For example see, T. P. Cheng and L. F. Li, Gauge Theoryof Elementary Particle Physics (Oxford, 1988); M. E. Pe-skin and D. V. Schroeder, An Introduction to QuantumField Theory (Westview Press, 1995).[2] M. Senami and K. Ito, Phys. Rev. A 99, 012509 (2019).[3] M. Senami, K. Inada, K. Soga, M. Fukuda, A. Tachibana,Prog. Theor. Chem. Phys. 31, 95 (2018).[4] R. Naaman, J. Phys. Chem. Lett. 3, 2178 (2012); R.Naaman, D. H. Waldeck, Annu. Rev. Phys. Chem. 66,263 (2015).[5] A. Tachibana, J. Mol. Struct.: (THEOCHEM), , 138(2010); T. Hara, M. Senami, A. Tachibana, Phys. Lett. A376, 1434 (2012); M. Fukuda, M. Senami, A. Tachibana,Prog. Theor. Chem. Phys. 27, 131(2013); M. Fukuda, K.Soga, M. Senami, A. Tachibana, Int. J. Quant. Chem116, 920 (2016).[6] A. Tachibana, New Aspects of Quantum Electrodynam-ics, (Springer 2017).[7] For example, see, U. Meierhenrich, “Amino Acids andthe Asymmetry of Life”, (Springer-Verlag, 2008).[8] F. Vester, T. L. V. Ulbricht, and H. Krauch, Natur-wissenschaften 46, 68 (1959); T. L. V. Ulbricht and F.Vester, Tetrahedron 18, 629 (1962).[9] For example see, J. D. Bjorken and S. D. Drell, Relativis-tic Quantum Mechanics (McGraw-Hill, 1964); C. Itzyk-son and J. B. Zuber, Quantum field theory (McGraw-Hill,1980).[10] R. A. Hegstrom, J. P. Chamberlain, K. Seto, R. G. Wat-son, Am. J. Phys. 56, 1086 (1988); R. A. Hegstrom, J.Mol. Struct. (Theochem), 232, 17 (1991).[11] R. Bast, A. Koers, A. S. P. Gomes, M. Iliaˇs, L. Visscher,P. Schwerdtfeger, T. Saue, Phys. Chem. Chem. Phys. 13,864 (2011).[12] DIRAC, a relativistic ab initio electronic structure pro- gram, Release DIRAC17 (2017), written by L. Viss-cher, H. J. Aa. Jensen, R. Bast, and T. Saue, withcontributions from V. Bakken, K. G. Dyall, S. Dubil-lard, U. Ekstr¨om, E. Eliav, T. Enevoldsen, E. Faßhauer,T. Fleig, O. Fossgaard, A. S. P. Gomes, E. D. Hedeg˚ard,T. Helgaker, J. Henriksson, M. Iliaˇs, Ch. R. Jacob,S. Knecht, S. Komorovsk´y, O. Kullie, J. K. Lærdahl,C. V. Larsen, Y. S. Lee, H. S. Nataraj, M. K. Nayak,P. Norman, G. Olejniczak, J. Olsen, J. M. H. Olsen,Y. C. Park, J. K. Pedersen, M. Pernpointner, R. di Remi-gio, K. Ruud, P. Sa lek, B. Schimmelpfennig, A. Shee,J. Sikkema, A. J. Thorvaldsen, J. Thyssen, J. van Stralen,S. Villaume, O. Visser, T. Winther, and S. Yamamoto(see )[13] QEDynamics , M. Senami, K. Ichikawa, A. Tachibana,(https://github.com/mfukudaQED/QEDalpha)[14] K.G. Dyall, Theor. Chem. Acc. 99, 366 (1998); Theor.Chem. Acc. 108, 335 (2002); Theor. Chem. Acc. 115,441 (2006); Theor. Chem. Acc. 117, 483 (2007); J. Phys.Chem. A 113, 12638 (2009); Theor. Chem. Acc. 131, 1172(2012); Theor. Chem. Acc. 131, 1217 (2012).[15] T. H. Dunning, J. Chem. Phys. 90, 1007 (1989); A. K.Wilson, T. van Mourik, T. H. Dunning, J. Mol. Struc.THEOCHEM, 388, 339 (1996);[16] K. A. Peterson, D. E. Woon, T. H. Dunning, J. Chem.Phys. 100, 7410 (1994).[17] R. A. Kendall, T. H. Dunning, R. J. Harrison, J. Chem.Phys. 96, 6796 (1992); T. van Mourik, A. K. Wilson, T.H. Dunning, Mol. Phys. 96, 529 (1999).[18] M. Tanabashi et al. (Particle Data Group), Phys. Rev.D 98, 030001 (2018).[19] A. J. MacDermott, T. Fu, G. O. Hyde, R. Nakatsuka, andA. P. Coleman, Orig. Life Evol. Biosp. 39, 407 (2009).
TABLE III: Each orbital contribution to the total electron chirality of alanine, serine, and valine for the result of ONC+q-dyall.ae3z. E i is the orbital energy, and E PV ,i is the orbital contribution to the parity-violating energy difference. The value ofthe orbital contribution to electron chirality is shown for one electron in Kramers pair, while both values in pairs are summedfor the total. Alanine Serine ValineNo. E i [a.u.] R d xψ † i γ ψ i No. E i [a.u.] R d xψ † i γ ψ i No. E i [a.u.] R d xψ † i γ ψ i − .
611 2 . × − − .
615 4 . × − − .
609 9 . × − − .
548 4 . × − − . − . × − − .
546 5 . × − − . − . × − − .
554 5 . × − − . − . × − − .
374 1 . × − − . − . × − − .
372 6 . × − − . − . × − − .
379 3 . × − − . − . × − − .
236 4 . × − − .
296 1 . × − − .
246 5 . × − − . − . × − − . − . × − − . − . × − − . − . × − − . − . × − − .
227 8 . × − − . − . × − − . − . × − − . − . × − − .
029 6 . × − − . − . × − − . − . × − − .
905 1 . × − − . − . × − − . − . × − − .
820 5 . × − − .
022 1 . × − − . − . × − − . − . × − − .
897 6 . × − − .
964 1 . × − − .
689 3 . × − − .
823 4 . × − − . − . × − − . − . × − − .
764 5 . × − − . − . × − − . − . × − − . − . × − − .
795 1 . × − − .
618 2 . × − − .
685 1 . × − − . − . × − − . − . × − − . − . × − − .
697 2 . × − − . − . × − − .
653 1 . × − − . − . × − − . − . × − − .
631 2 . × − − . − . × − − .
527 5 . × − − . − . × − − . − . × − − . − . × − − .
569 6 . × − − .
619 1 . × − − .
459 4 . × − − . − . × − − .
588 1 . × − − .
421 3 . × − − . − . × − − . − . × − − .
477 5 . × − − .
550 5 . × − − .
440 7 . × − − .
538 6 . × − − . − . × − − . − . × − − .
425 1 . × − − .
490 8 . × − − .
487 1 . × − − . − . × − − .
454 8 . × − − .
413 1 . × − Total 2 . × − . × − . × −−