EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-130 LHCb-PAPER-2014-023 September 10, 2014

First measurement of the charge asymmetry in beauty-quark pair production at a
hadron collider

The LHCb collaboration†††Authors are listed on the following pages.

The difference in the angular distributions between beauty quarks and
antiquarks, referred to as the charge asymmetry, is measured for the first
time in ${b}{\overline{b}}$ pair production at a hadron collider. The data
used correspond to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ collected
at 7$\mathrm{\,Te\kern-1.00006ptV}$ center-of-mass energy in proton-proton
collisions with the LHCb detector. The measurement is performed in three
regions of the invariant mass of the ${b}{\overline{b}}$ system. The results
obtained are

$\displaystyle A_{\rm
C}^{b\bar{b}}(40<M_{b\bar{b}}<75{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})$
$\displaystyle=$ $\displaystyle 0.4\pm 0.4\mathrm{\,(stat)}\pm
0.3\mathrm{\,(syst)}\%,$ $\displaystyle A_{\rm
C}^{b\bar{b}}(75<M_{b\bar{b}}<105{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})$
$\displaystyle=$ $\displaystyle 2.0\pm 0.9\mathrm{\,(stat)}\pm
0.6\mathrm{\,(syst)}\%,$ $\displaystyle A_{\rm
C}^{b\bar{b}}(M_{b\bar{b}}>105{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})$
$\displaystyle=$ $\displaystyle 1.6\pm 1.7\mathrm{\,(stat)}\pm
0.6\mathrm{\,(syst)}\%,$

where $A_{\rm C}^{b\bar{b}}$ is defined as the asymmetry in the difference in
rapidity between jets formed from the beauty quark and antiquark. The beauty
jets are required to satisfy $2<\eta<4$, $\mbox{$E_{\rm
T}$}>20\mathrm{\,Ge\kern-1.00006ptV}$, and have an opening angle in the
transverse plane $\Delta\phi>2.6$ rad. These measurements are consistent with
the predictions of the Standard Model.

© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.

LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, V.
Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1,
S. Belogurov31, K. Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S.
Benson38, J. Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van
Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M.
Bjørnstad54, T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A.
Bondar34, N. Bondar30,38, W. Bonivento15,38, S. Borghi54, A. Borgia59, M.
Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16, T. Brambach9, J. van den
Brand42, J. Bressieux39, D. Brett54, M. Britsch10, T. Britton59, J.
Brodzicka54, N.H. Brook46, H. Brown52, A. Bursche40, G. Busetto22,r, J.
Buytaert38, S. Cadeddu15, R. Calabrese16,f, M. Calvi20,k, M. Calvo Gomez36,p,
A. Camboni36, P. Campana18,38, D. Campora Perez38, A. Carbone14,d, G.
Carboni24,l, R. Cardinale19,38,j, A. Cardini15, H. Carranza-Mejia50, L.
Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20, L. Castillo Garcia38,
M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph. Charpentier38, S.
Chen54, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38, X. Cid
Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J.
Closier38, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A. Comerma-
Montells11, A. Contu15, A. Cook46, M. Coombes46, S. Coquereau8, G. Corti38, M.
Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De Capua54, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach55, O.
Deschamps5, F. Dettori38, A. Di Canto38, H. Dijkstra38, S. Donleavy52, F.
Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, K.
Dreimanis52, G. Dujany54, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S.
Eidelman34, S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I.
El Rifai5, Ch. Elsasser40, S. Ely59, S. Esen11, H.-M. Evans47, T. Evans55, A.
Falabella14, C. Färber11, C. Farinelli41, N. Farley45, S. Farry52, RF Fay52,
D. Ferguson50, V. Fernandez Albor37, F. Ferreira Rodrigues1, M. Ferro-Luzzi38,
S. Filippov33, M. Fiore16,f, M. Fiorini16,f, M. Firlej27, C. Fitzpatrick38, T.
Fiutowski27, M. Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M.
Frank38, C. Frei38, M. Frosini17,38,g, J. Fu21,38, E. Furfaro24,l, A. Gallas
Torreira37, D. Galli14,d, S. Gallorini22, S. Gambetta19,j, M. Gandelman2, P.
Gandini59, Y. Gao3, J. García Pardiñas37, J. Garofoli59, J. Garra Tico47, L.
Garrido36, C. Gaspar38, R. Gauld55, L. Gavardi9, G. Gavrilov30, E.
Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A. Gianelle22, S.
Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38, C. Göbel60, D.
Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, H. Gordon38, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E. Graugés36,
G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P. Griffith45, L.
Grillo11, O. Grünberg62, B. Gui59, E. Gushchin33, Yu. Guz35,38, T. Gys38, C.
Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S. Hall53, B.
Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N. Harnew55, S.T.
Harnew46, J. Harrison54, J. He38, T. Head38, V. Heijne41, K. Hennessy52, P.
Henrard5, L. Henry8, J.A. Hernando Morata37, E. van Herwijnen38, M. Heß62, A.
Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55,
N. Hussain55, D. Hutchcroft52, D. Hynds51, M. Idzik27, P. Ilten56, R.
Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, P. Jaton39, A. Jawahery58,
F. Jing3, M. John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N.
Jurik59, M. Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38,
S. Karodia51, M. Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C.
Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O. Kochebina7, M. Kolpin11,
I. Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, A.
Kozlinskiy41, L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P.
Krokovny34, F. Kruse9, W. Kucewicz26,o, M. Kucharczyk20,26,38,k, V.
Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38,
G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E. Lanciotti38, G.
Lanfranchi18, C. Langenbruch38, B. Langhans38, T. Latham48, C. Lazzeroni45, R.
Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32, J.
Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11, Y. Li3, M.
Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G. Liu38, S.
Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40, H. Lu3, D.
Lucchesi22,r, H. Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, F. Machefert7,
I.V. Machikhiliyan31, F. Maciuc29, O. Maev30, S. Malde55, G. Manca15,e, G.
Mancinelli6, J. Maratas5, J.F. Marchand4, U. Marconi14, C. Marin Benito36, P.
Marino23,t, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli41, D. Martinez Santos42, F. Martinez Vidal64, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57, F. Meier9, M. Meissner11, M. Merk41, D.A. Milanes8,
M.-N. Minard4, N. Moggi14, J. Molina Rodriguez60, S. Monteil5, M. Morandin22,
P. Morawski27, A. Mordà6, M.J. Morello23,t, J. Moron27, A.-B. Morris50, R.
Mountain59, F. Muheim50, K. Müller40, R. Muresan29, M. Mussini14, B. Muster39,
P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21,
S. Neubert38, N. Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C.
Nguyen-Mau39,q, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A.
Novoselov35, D.P. O’Hanlon48, A. Oblakowska-Mucha27, V. Obraztsov35, S.
Oggero41, S. Ogilvy51, O. Okhrimenko44, R. Oldeman15,e, G. Onderwater65, M.
Orlandea29, J.M. Otalora Goicochea2, P. Owen53, A. Oyanguren64, B.K. Pal59, A.
Palano13,c, F. Palombo21,u, M. Palutan18, J. Panman38, A. Papanestis49,38, M.
Pappagallo51, C. Parkes54, C.J. Parkinson9,45, G. Passaleva17, G.D. Patel52,
M. Patel53, C. Patrignani19,j, A. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez Trigo37, P.
Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen66, K. Petridis53, A.
Petrolini19,j, E. Picatoste Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25,
A. Pistone19, S. Playfer50, M. Plo Casasus37, F. Polci8, A. Poluektov48,34, E.
Polycarpo2, A. Popov35, D. Popov10, B. Popovici29, C. Potterat2, E. Price46,
J. Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig
Navarro39, G. Punzi23,s, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S. Ricciardi49, S.
Richards46, M. Rihl38, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5, P.
Robbe7, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V.
Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F.
Ruffini23, H. Ruiz36, P. Ruiz Valls64, G. Sabatino25,l, J.J. Saborido Silva37,
N. Sagidova30, P. Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez
Mayordomo64, B. Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37,
E. Santovetti24,l, M. Sapunov6, A. Sarti18,m, C. Satriano25,n, A. Satta24,
D.M. Saunders46, M. Savrie16,f, D. Savrina31,32, M. Schiller42, H.
Schindler38, M. Schlupp9, M. Schmelling10, B. Schmidt38, O. Schneider39, A.
Schopper38, M.-H. Schune7, R. Schwemmer38, B. Sciascia18, A. Sciubba25, M.
Seco37, A. Semennikov31, I. Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P.
Seyfert11, M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L.
Shekhtman34, V. Shevchenko63, A. Shires9, R. Silva Coutinho48, G. Simi22, M.
Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E.
Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51,
F. Soomro39, D. Souza46, B. Souza De Paula2, B. Spaan9, A. Sparkes50, P.
Spradlin51, F. Stagni38, M. Stahl11, S. Stahl11, O. Steinkamp40, O.
Stenyakin35, S. Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S.
Stracka23,38, M. Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38,
L. Sun57, W. Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M.
Szczekowski28, P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M.
Teklishyn7, G. Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg41, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38,
A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B.
Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,p, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H.
Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23,
S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P. Williams45, M.
Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, L. Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A.
Zhokhov31, L. Zhong3, A. Zvyagin38.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy

Measurements in $p\bar{p}$ collisions at the Tevatron [1, 2, 3, 4, 5, 6]
suggest that (anti)top quarks are produced along the (anti)proton beam
direction more often than predicted by the Standard Model (SM) [7]. Many
extensions to the SM have been proposed to explain this discrepancy (for a
review, see Ref. [8]) that couple new particles to quarks in a variety of
ways. Therefore, constraints on quark-antiquark production charge asymmetries
other than top-anti-top ($t\bar{t}$) could discriminate between models and be
used as a probe of non-SM physics. For example, some theories proposed to
explain the Tevatron results also predict a large charge asymmetry in
$b\bar{b}$ production [9, 10]. No measurement has been made to date of the
${b}{\overline{b}}$ charge asymmetry at a hadron collider.

The symmetric initial state of proton-proton collisions at the LHC does not
permit a charge asymmetry to be manifest as an observable defined using the
direction of one beam relative to the other. However, the asymmetry in the
momentum fraction of quarks and antiquarks inside the proton means that a
charge asymmetry can lead to a difference in the rapidity distributions of
beauty quarks and antiquarks. The $b\bar{b}$ charge asymmetry in $pp$
collisions is defined as

$A_{\rm C}^{b\bar{b}}\equiv\frac{N(\Delta y>0)-N(\Delta y<0)}{N(\Delta
y>0)+N(\Delta y<0)},$ (1)

where $\Delta y\equiv|y_{b}|-|y_{\bar{b}}|$ is the rapidity difference between
jets formed from the $b$ and $\bar{b}$ quarks. Measurements of the top-quark
charge asymmetry by the ATLAS and CMS experiments are consistent with the SM
expectations [11, 12, 13]. However, the large $gg\rightarrow t\bar{t}$ cross-
section at the LHC dilutes the observable signal of new physics entering the
$q\bar{q}\rightarrow t\bar{t}$ process that dominates $t\bar{t}$ production at
the Tevatron.

In the SM the only sizable leading-order (LO) contribution to $A_{\rm
C}^{b\bar{b}}$ comes from $Z\rightarrow{{b}{\overline{b}}}$ decays. The
contribution of $Z\rightarrow{{b}{\overline{b}}}$ to $A_{\rm C}^{b\bar{b}}$ in
a region of invariant mass of the ${b}{\overline{b}}$ system ($M_{b\bar{b}}$)
around the $Z$ boson mass is expected to be about 2% based on simulation.
Production of ${b}{\overline{b}}$ pairs at LO in quantum chromodynamics (QCD)
is symmetric under the exchange of $b$ and $\bar{b}$ quarks. At higher orders,
radiative corrections to the $q\bar{q}\rightarrow{{b}{\overline{b}}}$ process
generate an asymmetry in the differential distributions of the $b$ and
$\bar{b}$ quarks and induce a correlation between the direction of the $b$
($\bar{b}$) quark and that of the incoming $q$ ($\bar{q}$) quark. Such higher-
order corrections are expected to be negligible at low $M_{b\bar{b}}$ and to
increase in importance at larger $M_{b\bar{b}}$. The contribution to $A_{\rm
C}^{b\bar{b}}$ from higher-order terms is expected to reach 1% near the $Z$
boson mass [14]. Precision measurements of $A_{\rm C}^{b\bar{b}}$ as a
function of $M_{b\bar{b}}$ are sensitive probes of physics beyond the SM.

This Letter reports the first measurement of the charge asymmetry in beauty-
quark pair production at a hadron collider. The data used correspond to an
integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ collected at
7$\mathrm{\,Te\kern-1.00006ptV}$ center-of-mass energy in $pp$ collisions with
the LHCb detector. The measurement is performed in three regions of
$M_{b\bar{b}}$:
${40<M_{b\bar{b}}<75{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$,
${75<M_{b\bar{b}}<105{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$ and
${M_{b\bar{b}}>105{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$. This scheme is
chosen such that the middle region is centered around the mass of the $Z$
boson and contains most of the $Z\rightarrow{{b}{\overline{b}}}$ candidates.
The measurement is corrected to a pair of particle-level jets, each with a
pseudorapidity $2<\eta<4$, transverse energy $\mbox{$E_{\rm
T}$}>20\mathrm{\,Ge\kern-1.00006ptV}$, and an opening angle between the jets
in the transverse plane $\Delta\phi>2.6$ rad.

The LHCb detector is a single-arm forward spectrometer covering the range
$2<\eta<5$ designed for the study of particles containing $b$ or $c$ quarks,
described in detail in Refs.[15, 16, 17, 18]. The trigger [19] consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction.
Identification of beauty-hadron decays in the software trigger requires a
two-, three- or four-track secondary vertex with a large sum of the transverse
momentum ($p_{\rm T}$) of the tracks and a significant displacement from the
primary $pp$ interaction vertices. A multivariate algorithm [20] is used for
the identification of vertices consistent with the decay of a beauty hadron.
This so-called topological trigger algorithm (TOPO) is also used in this
analysis to identify the hadrons that contain the beauty quark and antiquark
in ${b}{\overline{b}}$ pair production. The charge of the beauty (anti)quarks
is determined by the charge of muons originating from semileptonic beauty-
hadron decays.

Simulated events are used to calibrate the jet energy scale, to determine the
reconstruction and selection efficiencies and to unfold the detector response.
In the simulation, $pp$ collisions are generated using Pythia [21] with a
specific LHCb configuration [22]. Decays of hadronic particles are described
by EvtGen [23], in which final state radiation is generated using Photos [24].
The interaction of particles with the detector and its response are
implemented using the Geant4 toolkit [25, *Agostinelli:2002hh] as described in
Ref. [27].

The ${b}{\overline{b}}$ are reconstructed as jets using the anti-$k_{\rm T}$
algorithm [28] with distance parameter $R=0.7$, as implemented in Fastjet
[29]. The inputs to the jet reconstruction are selected using a particle flow
approach [30]. Information from all the detector sub-systems is used to create
charged and neutral particle inputs to the jet algorithm. Jet-quality criteria
are applied to remove jets for which a large fraction of the energy is likely
due to sources other than a $pp$ collision, e.g., detector noise or poorly
reconstructed tracks. The per-jet efficiency of these criteria is $90-95\%$
depending on the jet kinematic properties. The mean number of $pp$ collisions
per event is only 1.8 making it unlikely to produce ${b}{\overline{b}}$ in
separate collisions; however, to prevent this, both jets are required to
originate from the same $pp$ collision.

The observed energy of each jet is corrected to the particle-level energy
accounting for the following effects: imperfect detector response; the
presence of detector noise; energy contributions from $pp$ interactions other
than the one in which the ${b}{\overline{b}}$ are produced; beauty (anti)quark
energy flowing out of the jet cone; and the presence of a neutrino from the
semileptonic decay of a beauty hadron in the jet. The jet energy correction
varies in the range $0-20\%(\pm 10\%)$ for jets that do(do not) contain a
neutrino from a semileptonic beauty-hadron decay. The mean value for jets that
do not contain a semileptonic-decay neutrino is about 1%. This correction is
obtained from simulation and depends on the jet $\eta$, $E_{\rm T}$, and the
number of $pp$ interactions in the event. Only jets in a well-understood
kinematic regime of LHCb, $\mbox{$E_{\rm T}$}>20\mathrm{\,Ge\kern-1.00006ptV}$
and $2<\eta<4$, are considered in this analysis. The relative resolution on
$M_{b\bar{b}}$ obtained using these jets is about 15%.

Jets in events selected by the TOPO need to be identified (tagged) as
containing a beauty quark or antiquark (bTAG). For this task, an association
is made between jets and the multitrack TOPO objects. If at least 60% of the
detector hits that make up the tracks forming the TOPO object also belong to
tracks within the jet, then the jet satisfies a bTAG requirement. At least one
jet in the event is required to contain a beauty-hadron decay selected by the
TOPO which caused the event to be recorded. The TOPO is applied to offline-
reconstructed tracks with a looser requirement to search for a second beauty-
hadron decay in the event. If such a decay is found, and if it can be
associated to another jet, then the event is identified as containing a
${b}{\overline{b}}$ pair. The mean di-bTAG efficiency for di-jet events used
in this analysis is about 30%, while the per-jet mis-tag efficiency for jets
initiated by light quarks and gluons is less than 0.1%. To enhance the
contribution of non-$gg$ production mechanisms, $\Delta\phi>2.6~{}\rm\,rad$ is
required between the two jets that satisfy the bTAG requirement.

The largest background contribution is due to charm jets. The level of
background contamination is determined using the so-called corrected mass

$M_{\rm
corr}=\sqrt{M^{2}+\left(\frac{p}{c}\right)^{2}\sin^{2}{\theta}}+\frac{p}{c}\sin{\theta},$
(2)

where $M$ and $p$ are the invariant mass and momentum of all tracks in the jet
that are inconsistent with originating directly from a $pp$ collision and have
a minimum distance of closest approach to a track used in the TOPO less than
0.2$\rm\,mm$. The angle $\theta$ is between the momentum and the direction
from the $pp$ collision to the TOPO object vertex. The corrected mass is the
minimum mass the long-lived hadron can have that is consistent with the
direction of flight.

Figure 1 shows the corrected-mass distribution. The corrected-mass probability
density functions (PDFs) for beauty and charm are obtained from simulation.
Imperfect measurement of the direction of flight can result in a larger
corrected mass than the true hadron mass. For charm-hadron decays, the
particles originate from a single point in space and typically the missing
momentum is carried by a single low-mass particle; thus, the corrected mass
peaks near the known charm-meson mass. The vast majority of beauty-hadron
decays involve intermediate charm hadrons which results in not all stable
particles originating from the same spatial point. The missing momentum is
typically carried away by multiple particles and the invariant mass of the
missing momentum may be large. Hence, the corrected mass for beauty decays
peaks below the known beauty-meson mass and has worse resolution than for
charm. The result of a fit to the data shown in Fig. 1 is that $3.6\pm 1.2\%$
of events in the final sample are not ${b}{\overline{b}}$, where the
uncertainty is due to the corrected-mass PDFs. The contribution from jets
initiated by light quarks or gluons is found to be negligible. Furthermore,
the limited acceptance of the LHCb detector for $b\bar{b}$ originating from
$t\bar{t}$ makes this contribution negligible as well.

Figure 1: (top) Corrected mass of TOPO objects associated to bTAG jets in the
final event sample. Less than 2% of jets are found to originate from charm.
(bottom) Corrected mass of TOPO objects associated to sub-leading vs leading
jets in the final event sample. A small $c\bar{c}$ contribution is visible
near (2,2)${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

To measure the charge asymmetry, the charge of the beauty (anti)quark needs to
be identified in at least one of the jets (qTAG). The qTAG requirement is that
a track in the TOPO object and in the jet is identified as a muon. The muon is
required to satisfy $\mbox{$p_{\rm T}$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $p>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to reduce the charge asymmetry
due to detector biases. This strategy is designed to look for muons coming
from semileptonic beauty-hadron decays; thus, the charge of the muon tags the
charge of the beauty quark or antiquark. Decays of the type $b\rightarrow
c\rightarrow\mu$ contaminate the charge tagging. To mitigate this, the tagging
muon is required to have the highest momentum of all displaced tracks in the
jet. A further dilution to the charge-tagging purity arises due to
oscillations of the ${B}^{0}$ and ${B}^{0}_{s}$ mesons.

The expected qTAG purity, defined as the probability to correctly assign the
charge of the beauty quark in a qTAG jet, can be estimated using the
following: the measured $b$-hadron production fractions [31, 32]; the
$b$-hadron and $c$-hadron semileptonic branching fractions [33]; the charge-
tagging efficiencies for $b$ and $c$-hadron semileptonic decays obtained from
simulation; the ${B}^{0}$ and ${B}^{0}_{s}$ oscillation frequencies [34, 35]
and the reconstruction efficiency as a function of $b$-hadron lifetime
obtained from simulation. Combining all of this information yields an expected
qTAG purity of $73\pm 4\%$. The purity is expected to decrease by a few
percent with increasing jet energy due to an increase in the neutral-beauty-
meson production fractions.

The qTAG purity is measured directly using events where both bTAG jets also
satisfy the qTAG requirement using the fraction of events where the two muons
have opposite charges. This gives an integrated qTAG purity of $70.3\pm
0.3\%$, which agrees with the predicted value, and values of $71.6\pm 0.5\%$,
$68.8\pm 0.8\%$ and $66.1\pm 1.9\%$ for
$40<M_{b\bar{b}}<75{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
$75<M_{b\bar{b}}<105{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$M_{b\bar{b}}>105{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, respectively. The
observed decrease in purity agrees with expectations. The qTAG purity is found
to be consistent in data for all $\Delta y$. As a further consistency check, a
separate study of the qTAG purity is performed using events with a jet and a
fully reconstructed self-tagging $B^{+}\rightarrow J/\psi K^{+}$ or
$B^{+}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\pi^{+}$ decay. In these events,
the charge of the ${{B}^{+}}$ provides an unambiguous qTAG of the beauty jet
for ${b}{\overline{b}}$ pair production. Using $B^{+}+$jet events where the
jet satisfies the qTAG, the qTAG purity is determined to be $73\pm 3\%$. This
result agrees with both the predicted and di-qTAG results. The di-qTAG purity
measurement is used to obtain the final $A_{\rm C}^{b\bar{b}}$ results below.

Figure 2: Reconstructed $\Delta y$ distribution for all selected events after
background subtraction and correction for qTAG impurity. The dashed line shows
the distribution reflected about the vertical axis.

Figure 2 shows the $\Delta y$ distribution after background subtraction and
correcting for qTAG impurity. The reconstructed distributions of $\Delta y$
and $M_{b\bar{b}}$ are corrected for the effects of detector resolution and
for event reconstruction and selection efficiency. The correction for detector
resolution is achieved by applying a two-dimensional unfolding procedure to
the data [36]. The migration matrix in $\Delta y$ and $M_{b\bar{b}}$ is shown
in Fig. 3. The selection efficiency is obtained from simulated events as a
function of $\Delta y$ and $M_{b\bar{b}}$. The residual dependence of the
efficiency on other jet kinematic variables has a negligible impact on the
resulting measurement of $A_{\rm C}^{b\bar{b}}$.

Figure 3: Migration matrix in $\Delta y$ and $M_{b\bar{b}}$.

The main sources of systematic uncertainties on the measurement of $A_{\rm
C}^{b\bar{b}}$ are as follows: precision of the qTAG purity and its dependence
on jet kinematic properties; uncertainty in the unfolding; determination of
the selection efficiency; and any residual detector-related asymmetries. Table
1 summarizes the values of the systematic uncertainties assigned to the
measurement of $A_{\rm C}^{b\bar{b}}$ in each $M_{b\bar{b}}$ region.
Measurement of the qTAG purity is data-driven and the statistical
uncertainties are propagated to $A_{\rm C}^{b\bar{b}}$ to determine the
systematic uncertainty. The uncertainty due to unfolding accounts for the
choice of data sample used to generate the migration matrix and mismodeling of
the detector response in the simulation. The uncertainty due to efficiency is
dominated by the statistical uncertainty of the simulation. The polarity of
the LHCb dipole magnet is reversed periodically. This coupled with the hard
momentum spectrum of the tagging muons results in only small detection-based
asymmetries. Additionally, due to the definition of $\Delta y$, these
detection asymmetries cancel to very good approximation when summing over
$\mu^{+}$ and $\mu^{-}$ tags. The detection asymmetry of charged kaons causes
a negligible bias in $A_{\rm C}^{b\bar{b}}$.

Figure 4: Corrected $\Delta y$ distribution for all selected events. The
statistical uncertainties are negligible. The systematic uncertainties are
highly correlated from bin to bin and largely cancel in the determination of
$A_{\rm C}^{b\bar{b}}$. The LO SM prediction obtained from Pythia [37, 38] is
also shown.

Figure 4 shows the corrected $\Delta y$ distribution summed over all
$M_{b\bar{b}}$ regions considered
($M_{b\bar{b}}>40{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$). The LO SM
prediction, which includes LO QCD and $Z\rightarrow b\bar{b}$, obtained from
Pythia [37, 38] is also shown. The SM uncertainty includes contributions from
the renormalization and factorization scales, and from the parton distribution
functions. A next-to-LO SM calculation is required to obtain $A_{\rm
C}^{b\bar{b}}$ at the percent level. However, the LO result is sufficient to
demonstrate agreement between the theory and unfolded ${b}{\overline{b}}$
pair-production distribution.

The measurement of $A_{\rm C}^{b\bar{b}}$ is performed in three regions of
$M_{b\bar{b}}$ and the results obtained are

$\displaystyle A_{\rm C}^{b\bar{b}}(40,75)$ $\displaystyle=$ $\displaystyle
0.4\pm 0.4\mathrm{\,(stat)}\pm 0.3\mathrm{\,(syst)}\%,$ $\displaystyle A_{\rm
C}^{b\bar{b}}(75,105)$ $\displaystyle=$ $\displaystyle 2.0\pm
0.9\mathrm{\,(stat)}\pm 0.6\mathrm{\,(syst)}\%,$ $\displaystyle A_{\rm
C}^{b\bar{b}}(>105)$ $\displaystyle=$ $\displaystyle 1.6\pm
1.7\mathrm{\,(stat)}\pm 0.6\mathrm{\,(syst)}\%,$

where the ranges denote the regions of $M_{b\bar{b}}$ in units of
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. These measurements are the first
to date of the charge asymmetry in ${b}{\overline{b}}$ pair production at a
hadron collider. The results are corrected to a pair of particle-level jets
each with $2<\eta<4$, $\mbox{$E_{\rm T}$}>20\mathrm{\,Ge\kern-1.00006ptV}$ and
$\Delta\phi>2.6$ rad between the jets. All results are consistent with the SM
expectations.

Table 1: Absolute systematic uncertainties (%). |  $M_{b\bar{b}}$ (GeV/$c^{2}$)
---|---
Source | $(40,75)$ | $(75,105)$ | $>105$
Mis-qTAG | $-$ | 0.1 | 0.2
Unfolding | 0.3 | 0.6 | 0.4
$\varepsilon(M_{b\bar{b}},\Delta y)$ | 0.1 | 0.1 | 0.2
$\varepsilon(b)-\varepsilon(\bar{b})$ | $-$ | $-$ | 0.3
Total | 0.3 | 0.6 | 0.6

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland);
INFN (Italy); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA
(Romania); MinES and FANO (Russia); MinECo (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia). Individual groups or members have received support from EPLANET,
Marie Skłodowska-Curie Actions and ERC (European Union), Conseil général de
Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR
(Russia), XuntaGal and GENCAT (Spain), Royal Society and Royal Commission for
the Exhibition of 1851 (United Kingdom).

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# Towards Complementary Characterization
of the Chemical Bond

Maciej Hendzel  [  Maciej Fidrysiak  Józef Spałek<EMAIL_ADDRESS>
###### Abstract

A precise discussion of a single bond requires consideration of two-particle
wave function for the particles involved. Here we define and determine
rigorously the intrinsic covalency and connected characteristics on the
canonical example of H2 molecule. This is achieved by starting from analytic
form for the two–particle wave function for electrons forming the bond, in
which we single out the atomic contribution (atomicity) in an unequivocal
manner. The presence the of atomicity and ionicity factors complements the
existing attributes of the bond. In this way, a gradual evolution of the
molecular state to its two-atomic correspondant is traced systematically with
increasing interatomic distance. In effect, a direct relation to the onset of
incipient Mott-Hubbard atomicity (Mottness) to the intrinsic covalency and
ionicity is established. This goal is achieved by combining the
single–particle wave function readjustment with a simultaneous determination
of two–particle states in the particle (second–quantization) representation.

font=sf,small Jagiellonian University] Institute of Theoretical Physics,
Jagiellonian University, ulica Łojasiewicza 11,
PL-30-348 Kraków, Poland

The concept of chemical bond and its quantum properties are of fundamental
importance to our understanding of both physical and chemical characteristics
of molecules and solid state compounds 1, 2, 3. Among the principal questions
are those of relative role of covalency, ionicity, and atomicity, as they
describe qualitative differences with the characteristics of parent atomic
states composing the system. Here we propose a resolution of the question
concerning the evolution of the molecular into corresponding atomic states and
vice versa. In effect, such an approach leads to an unequivocal determination
of true covalency, as well as to extracting both the atomicity and ionicity.
As a side result, we resolve the longstanding question of unphysical behavior
of covalency with the increasing interatomic distance (see also Supporting
Information).

At the outset, we take a multiparticle view of the chemical bond and implement
a special method EDABI (Exact Diagonalization Ab Initio approach) devised in
our group earlier 4, 5, 6 and apply it here to a rigorous analysis in the
simplest situation of the H2 molecule. The generic case of the H2 molecule
involves two–electron single bond and thus the role of interelectronic
correlations in combination with concomitant single-particle wave function
readjustment in the resultant (correlated) state, can be analyzed rigorously
and in mathematically analytic terms.

The structure of this Letter is as follows. After introducing exact two-
particle electron wave function we redefine the covalency to extract from
standard definition the contribution of atomicity, by referring to the notions
of Mott and Hubbard localization. This new formulation allows us to define and
determine explicitly the true covalency, atomicity (also termed seniority 7),
and ionicity factors in the chemical bond and hence to understand those
component factors of the chemical bond in a fully quantitative manner. The
whole methodology is based on combination of both the first– and
second–quantization aspects of the relevant multielectron states, detailed in
the Supporting Information.

By applying the procedure outlined in the Method and in Supporting Information
we obtain two–particle wave function in an explicit form

$\displaystyle\begin{split}&\Psi_{0}(\textbf{r}_{1},\textbf{r}_{2})=\frac{2(t+V)}{\sqrt{2D(D-U+K)}}\Psi_{cov}(\textbf{r}_{1},\textbf{r}_{2})-\frac{1}{2}\sqrt{\frac{D-U+K}{2D}}\Psi_{ion}(\textbf{r}_{1},\textbf{r}_{2})\equiv\\\
&\equiv
C\psi_{cov}(\textbf{r}_{1},\textbf{r}_{2})+I\psi_{ion}(\textbf{r}_{1},\textbf{r}_{2}),\end{split}$
(1)

where the covalent ($\Psi_{cov}$) and ionic ($\Psi_{ion}$) parts are

$\displaystyle\Psi_{cov}(\textbf{r}_{1},\textbf{r}_{2})=\left[w_{1}(\textbf{r}_{1})w_{2}(\textbf{r}_{2})+w_{1}(\textbf{r}_{2})w_{2}(\textbf{r}_{1})\right]$
$\displaystyle\times\left[\chi_{\uparrow}(\textbf{r}_{1})\chi_{\downarrow}(\textbf{r}_{2})-\chi_{\downarrow}(\textbf{r}_{1})\chi_{\uparrow}(\textbf{r}_{2})\right],$
(2)
$\displaystyle\Psi_{ion}(\textbf{r}_{1},\textbf{r}_{2})=\left[w_{1}(\textbf{r}_{1})w_{1}(\textbf{r}_{2})+w_{2}(\textbf{r}_{1})w_{2}(\textbf{r}_{2})\right]$
$\displaystyle\times\left[\chi_{\uparrow}(\textbf{r}_{1})\chi_{\downarrow}(\textbf{r}_{2})-\chi_{\downarrow}(\textbf{r}_{1})\chi_{\uparrow}(\textbf{r}_{2})\right],$
(3)

where $\\{w_{i}(\textbf{r})\\}_{i=1,2}$ are trial Hückel-Slater molecular
orbitals and $\\{\chi_{\sigma}(i)\\}_{\sigma=\pm 1}$ are spin functions. In
above expression the microscopic parameters, $U$ and $K$, are the magnitudes
of intraatomic and interatomic Coulomb repulsion, respectively, $t$ and $V$
are the magnitudes of hopping (Bloch integral) and so–called correlated
hopping, respectively (for details see Methods) and

$\displaystyle D\equiv\sqrt{(U-K)^{2}+16(t+V)^{2}}.$ (4)

We note that the two–particle wave function has the Heitler–London form,
except the coefficients contain all interparticle interaction terms and the
single–particle wave functions are the molecular Hückel-Slater orbitals with
an adjusted size in the resultant correlated state. This two–particle wave
function may be rewritten in terms of the original Slater orbitals as follows

$\displaystyle\begin{split}&\Psi_{0}(\textbf{r}_{1},\textbf{r}_{2})=\left(C\beta^{2}(1+\gamma^{2})-2\gamma
I\beta^{2}\right)\phi^{at}_{cov}(\textbf{r}_{1},\textbf{r}_{2})+\left(I\beta^{2}(1-\gamma^{2})-2\gamma
C\beta^{2}\right)\phi^{at}_{ion}(\textbf{r}_{1},\textbf{r}_{2})\\\
&\equiv\tilde{C}\phi^{at}_{cov}+\tilde{I}\phi^{at}_{ion},\end{split}$ (5)

where the coefficients $\beta$ and $\gamma$ are defined through the relation

$\displaystyle
w_{i\sigma}(\textbf{r})=\beta[\psi_{i\sigma}(\textbf{r})-\gamma\psi_{j\sigma}(\textbf{r})],$
(6)

where $i$ and $j$ label the atoms, $\sigma\equiv\pm
1\equiv\uparrow,\downarrow$ is the electron spin quantum number. Note that
$\braket{w_{i}(\textbf{r})}{w_{j}(\textbf{r})}=\delta_{ij}$, $\beta$ and
$\gamma$ are the mixing coefficients of the neighboring Slater orbitals
($\psi_{i\sigma}\equiv\psi(\textbf{r}-\textbf{R}_{i})\chi_{\sigma}(i)\equiv(\sqrt{\alpha^{3}/\pi})exp(-\alpha|\textbf{r}-\textbf{R}_{i}|)\chi_{\sigma}(i)$),
(in which $\alpha^{-1}$ is the size of the orbital). The functions
$\phi^{at}_{cov}(\textbf{r}_{1},\textbf{r}_{2})$ and
$\phi^{at}_{ion}(\textbf{r}_{1},\textbf{r}_{2})$ have the same form as (2) and
(3), respectively, except for the replacement
$w_{i\sigma}(\textbf{r})\rightarrow\psi_{i\sigma}(\textbf{r})$.

We define the effective atomic contribution $\tilde{A}$ at given interatomic
distance $R$ as

$\displaystyle lim_{\gamma\rightarrow 0}\tilde{C}=C\beta^{2}\equiv\tilde{A},$
(7)

i.e., regard it as the weight of atomic part at the same $R$. Formally, the
true covalency, ionicity, and atomicity are defined for given $R$ as

$\displaystyle\text{covalency:
}\gamma_{cov}\equiv\frac{|\tilde{C}|^{2}-|\tilde{A}|^{2}}{|\tilde{C}|^{2}+|\tilde{I}|^{2}},$
(8) $\displaystyle\text{ionicity:
}\gamma_{ion}\equiv\frac{|\tilde{I}|^{2}}{|\tilde{C}|^{2}+|\tilde{I}|^{2}},$
(9) $\displaystyle\text{atomicity:
}\gamma_{at}\equiv\frac{|\tilde{A}|^{2}}{|\tilde{C}|^{2}+|\tilde{I}|^{2}}.$
(10)

The original motivation for introducing the concept of atomicity is as
follows: It may seem natural to define covalency as
$|C|^{2}/(|C|^{2}+|I|^{2})$ and ionicity as $|I|^{2}/(|C|^{2}+|I|^{2})$.
However, such a choice leads directly to unphysical features (see Fig. 7 in
Supporting Information). Namely, $|C|^{2}/(|C|^{2}+|I|^{2})$ reaches its
maximal value of unity in the limit of separate atoms ($R\rightarrow\infty$).
This is also the limit when electrons are entirely localized on their parent
atoms and become distinguishable in the quantum mechanical sense. This is the
limit which we regard as that atomicity equal to unity and vanishing true
covalency. This type of argument is also the reason of subtracting the
probability $\tilde{A}^{2}$ from $\tilde{C}^{2}$, not the corresponding
wave–function amplitudes. The quantities (8)–(10) are proved next to be useful
and of crucial importance.

To demonstrate the fundamental meaning of the introduced quantities we relate
them to the criteria of Mott8 and Hubbard9 for onsets of localized (atomic)
behavior. This is because the evolution of molecular H2 (electron–paired)
state into individual separate singly occupied (atomic) states is regarded as
equivalent to the Mott–Hubbard localization (for recent related discussion in
different context see e.g. 10). Namely, we define Mott and Hubbard onset
criteria as

$\displaystyle\frac{2|t+V|}{U-K}=1,\text{ and
}n^{1/d}_{c}\alpha^{-1}\equiv\frac{1}{\alpha_{0}R_{Mott}}\simeq 0.5,$ (11)

Figure 1: Hubbard (orange) and Mott (blue) characteristics of atomicity vs.
interatomic distance $R$ (dashed horizontal lines). The dots mark the points
corresponding to Hubbard and Mott criteria. The vertical dotted line marks the
onset of Mottness at $R_{Mott}$. The inset: $R$ dependence of the orbital size
of the renormalized atomic wave functions composing the molecular (Wannier)
single-particle states. The dotted line marks the equilibrium distance
$R_{bond}$.

respectively, where $\alpha_{0}$ is the readjusted inverse orbital size, here
at $R=R_{Mott}$. The first of them implies that for $R=R_{Mott}$ the kinetic
(hopping) energy is equal to the correlation energy, i.e., for $R<R_{Mott}$
the ratio is greater than unity, whereas for $R>R_{Mott}$ it is smaller than
unity and reduces quite rapidly to zero with increasing $R$ beyond $R_{Mott}$.
In other words, the kinetic energy dominates in the former case and enhances
hopping electrons to resonate strongly between the sites, whereas the
electrons become gradually frozen as $R$ increases beyond $R_{Mott}$. On the
other hand, the Mott criterion expresses the onset of localization in terms of
the renormalized single–particle wave function size at the localization
threshold. Namely, the threshold is reached when the diameter of the orbital
in the correlated state ($2\alpha_{0}^{-1}$) is equal to the interorbital
distance ($R=R_{Mott}$). Semiclassically, it means that the collective
character is established when the orbitals start overlapping.

Figure 2: Intrinsic properties of the chemical bond: atomicity (green), true
covalency (blue), and ionicity (red), all as a function of interatomic
distance $R$. They represent the relative weights in the total two-particle
wave function. In the $R\rightarrow 0$ limit the atomicity practically
disappears and is the only contribution in the separate–atom limit
$R\rightarrow\infty$. The solid circle defines the onset of localization
effects (Mottness) due to interelectronic correlations. If the atomicity is
disregarded, the covalency exhibits a drastic nonphysical behavior with
increasing $R>R_{bond}$. The figure illustrates a systematic evolution of
molecular states into separate atoms and vice versa, formation of molecular
states out of separate atoms. The Slater states have a renormalized size
$\alpha^{-1}\leq\alpha_{B}$.

To visualize this formal reasoning we have plotted in Fig. 1 the left parts of
both (11) as a function of $R$, as well as have marked by red points their
values at $R_{Mott}$. The blue shaded area may be called the Mottness regime
11. Inset shows the corresponding dependence of the renormalized size of the
Slater orbital in the correlated state, with the dotted vertical line marking
the equilibrium bond length $R_{bond}\simeq 1.43a_{0}$. Note that the orbital
size for $R>R_{Mott}$ approaches rapidly to the free–atom values $a_{0}$.

Figure 3: Electron density $n_{\sigma}(\textbf{r})$ for different interatomic
distance (a) $R=1a_{0}^{-1}$, (b) $R=1.43a_{0}^{-1}$, (c) $R=2.3a_{0}^{-1}$,
(d) $R=4a_{0}^{-1}$. The parts centered at nuclei are practically disjoint for
$R\geq 5a_{0}$, illustrating the robustness of atomic behavior in that
situation. This density contains also the double-occupancy (ionicity)
contribution which becomes rapidly negligible with increasing $R$ beyond
$R_{Mott}$.

The explicit connection of the above onset to the true covalency
($\gamma_{cov}$), ionicity ($\gamma_{ion}$), and atomicity ($\gamma_{at}$) is
visualized in Fig. 2, where the $R$ dependence of those quantities is drawn.
Remarkably, at the distance $R_{Mott}$ the true covalency and atomicity
acquire the same value, so the point $R_{Mott}$ is a crossover point from
true–covalency dominated to atomicity (Mottness) regime. Furthermore,
$\gamma_{cov}$ is predominant for $R<R_{Mott}$, whereas $\gamma_{at}$ is for
$R>R_{Mott}$, as it should. Additionally, the covalency and ionicity (atom
double occupancy) coincide as $R\rightarrow 0$, whereas then
$\gamma_{at}\rightarrow 0$. Both $\gamma_{cov}$ and $\gamma_{ion}$ disappear
in the atomic limit ($R>>R_{Mott}$), where $\gamma_{at}\rightarrow 1$. The
results presented in Fig. 2 illustrate one the central findings of the present
work. The principal characteristics are detailed further in the Tables 1 and
2. In Table 1 we list the discussed factors of Mottness onset to show that
they are mutually consistent. This agreement leads to the conclusion that the
introduced entities (8)–(10) are not only relevant for the description of the
Mott–Hubbard localization in condensed-matter (extended) systems, but also
appear as a crucial incipient feature in molecular systems. We stress, this
was possible only by introducing two-particle wave function as the proper
characteristic of single bond which, after all, is composed of electron pairs.

Table 1: Equivalent characteristics of the atomicity onset threshold (for details see main text). Characteristic | Value
---|---
Mott criterion | 0.42
Hubbard criterion | 1
Covalent-atomic crossover | 2.285$a_{0}$
$R_{Mott}$ | 2.279$a_{0}$

R ($a_{0}$) | $n(0,0)$ | $\alpha_{0}$ | $\beta$ | $\gamma$
---|---|---|---|---
1 | 0.334 | 1.30751 | 1.1386 | 0.47811
1.43 | 0.267 | 1.19838 | 1.0854 | 0.38877
2.3 | 0.142 | 1.05428 | 1.0338 | 0.25358
4 | 0.023 | 0.998601 | 1.0012 | 0.048640

Table 2: Particles density at the mid point, inverse orbital size
$\alpha_{0}$, and mixing coefficients $\beta$ and $\gamma$, all versus $R$.
The particle density $n(0,0)$ illustrates the gradually vanishing electron
density in the region between the atoms as $R$ increases beyond $R_{Mott}$.

So far, our discussion was based on the wave–function mechanics. In the
remaining part we reformulate the analysis directly in the second–quantization
language which will allow us to provide the physical interpretation of the
bond in terms of particle densities. Namely, to amplify our multiparticle bond
description we return to the particle language and display in Fig. 3 several
panels composed of electron density in the $(x,y)$ plane, with the protons
distant by $R/a_{0}=1,$ $1.43$($R_{bond}$), $2.3$($R_{Mott}$), and $4$
(profiles a) - d)), respectively. The density is defined as

$\displaystyle
n_{\sigma}(\textbf{r})=\bra{\psi_{G}}\hat{\psi}^{\dagger}_{\sigma}(\textbf{r})\hat{\psi}_{\sigma}(\textbf{r})\ket{\psi_{G}},$
(12)

where $\ket{\psi_{G}}$ is the lowest spin-singlet eigenstate and
$\hat{\psi}_{\sigma}(\textbf{r})$ is the field operator. One should note that
this density when integrated and summed over spin directions ($\sigma\pm 1$)
is equal to the total number of particles ($N_{e}=2$). Obviously,
$n_{\uparrow}(\textbf{r})=n_{\downarrow}(\textbf{r})\equiv n(\textbf{r})$/2,
where $n(\textbf{r})$ is the total density. More importantly, this quantity
provides the physical density, in contrast to the probability density
$|\psi_{G}(\textbf{r}_{1},\textbf{r}_{2})|^{2}$. This distinctive feature of
$n(\textbf{r})$ shows that the density diminishes in the region between the
atoms to zero relatively fast with the increasing $R$ above $R_{Mott}$. To
substantiate the last statement we have listed in Tab. 2 the density value
$n(0,0)$ in the middle point between the proton positions. For the sake of
comparison, we have also added there the inverse orbital size, as well as the
mixing coefficients in the wave function
$w_{i}(\textbf{r})=w(\textbf{r}-\textbf{R}_{i})$ to show, that indeed both
two– and single–particle characteristics merge into their atomic
correspondants as $R$ increases beyond $R_{Mott}$. Note that whereas $\gamma$
expresses the decreasing Pauling covalency 12, $n(0,0)$ describes the
diminishing true covalency.

Analogously, we can define the concentration of the electrons in ionic
configuration as follows:

$\displaystyle
n_{ion}(\textbf{r})\equiv\bra{\psi_{G}}\hat{n}_{\sigma}(\textbf{r})\hat{n}_{\bar{\sigma}}(\textbf{r})\ket{\psi_{G}}$
(13)

with
$\hat{n}_{\sigma}(\textbf{r})\equiv\hat{\psi}^{\dagger}_{\sigma}(\textbf{r})\hat{\psi}_{\bar{\sigma}}(\textbf{r})$
and $\bar{\sigma}\equiv-\sigma$. In other words, $n_{ion}(\textbf{r})$
expresses the density of local spin-singlet pairs (double-site occupancy).
Profiles of $n_{ion}(\textbf{r})$ are not presented explicitly as they are
contained in the equivalent form as the second part of the wave function (5).

Finally, along with the definitions (12) and (13) of local particle densities,
we can define nonlocal density of covalent electrons in the following manner

$\displaystyle
n_{cov}(\textbf{r}_{1},\textbf{r}_{2})=\bra{\psi_{G}}\hat{n}_{\sigma}(\textbf{r}_{1})\hat{n}_{\bar{\sigma}}(\textbf{r}_{2})\ket{\psi_{G}}.$
(14)

This expression for correlation function completes our description in both the
first and second–quantization schemes. The expressions (12)–(14) may be useful
in the situations with more involved orbitals.

To summarize, our fairly complete analysis of single chemical bond is based
entirely on the multiparticle description, both in the first and
second–quantization schemes. Both of the descriptions are equivalent, but
within the second of them it is possible to relate it directly to the particle
language. In general, the approach may represent a starting point to a precise
experimental resolution of the bond factors, as well as to quantify their
specific features in more complex situations. Also, the method bridges the
atomic and molecular aspects of the chemical bond in precise multiparticle
categories. Analogous analysis may be carried out for the resolution of
covalent bonds evolution into ionic bonds and vice versa 13.

##  1 METHOD

Our analysis starts from full form of Hamiltonian in second quantization, with
all interaction terms between electrons on the lowest orbitals, i.e.,

$\displaystyle\mathcal{\hat{H}}=$
$\displaystyle\epsilon_{a}\sum_{i}\hat{n}_{i\sigma}+{\sum_{ij\sigma}}^{\prime}t_{ij}\,\hat{a}^{{\dagger}}_{i\sigma}\,\hat{a}_{j\sigma}+U\sum_{i}\hat{n}_{i\uparrow}\,\hat{n}_{i\downarrow}+\frac{1}{2}{\sum_{ij}}^{\prime}K_{ij}\hat{n}_{i\sigma}\,\hat{n}_{j\sigma^{\prime}}$
$\displaystyle-\frac{1}{2}{\sum_{ij}}^{\prime}J^{H}_{ij}\left(\hat{\textbf{S}}_{i}\cdot\hat{\textbf{S}}_{j}-\frac{1}{4}\hat{n}_{i}\hat{n}_{j}\right)+\frac{1}{2}{\sum_{ij}}^{\prime}{J}^{\prime}_{ij}(\hat{a}^{\dagger}_{i\uparrow}\hat{a}^{\dagger}_{i\downarrow}\hat{a}_{j\downarrow}\hat{a}_{j\uparrow}+\mathrm{H.c.})$
$\displaystyle+\frac{1}{2}{\sum_{ij}}^{\prime}V_{ij}(\hat{n}_{i\sigma}+\hat{n}_{j\sigma})(\hat{a}^{\dagger}_{i\bar{\sigma}}\hat{a}_{j\bar{\sigma}}+\mathrm{H.c.})+\mathcal{H}_{\text{ion-
ion}},$ (15)

Figure 4: Flowchart of the EDABI method. The method is initialized by
selection of a trial single-particle basis of wave functions (6), and
subsequent diagonalization of the many-particle Hamiltonian (15). Optimization
of the single-particle-state size leads to an explicit determination of the
trial-wavefunction parameters, microscopic interaction and hopping parameters
as well as ground-state energy, and explicit form of the many-particle
wavefunction, all in the correlated interacting state for a given interatomic
distance.

where $H.c$. denotes the Hermitian conjugation, $\hat{a}_{i\sigma}$
($\hat{a}^{\dagger}_{i\sigma}$) are fermionic annihilation (creation)
operators for state $i$ and spin $\sigma$,
$\hat{n}_{i\sigma}\equiv\hat{a}^{\dagger}_{i\sigma}\hat{a}_{i\sigma}$, and
$\hat{n}_{i}\equiv\hat{n}_{i\uparrow}+\hat{n}_{i\downarrow}\equiv\hat{n}_{i\sigma}+\hat{n}_{i\bar{\sigma}}$.
The spin operators are defined as
$\hat{S}_{i}\equiv\frac{1}{2}\sum_{\alpha\beta}\hat{a}^{\dagger}_{i\alpha}\sigma_{i}^{\alpha\beta}\hat{a}_{i\beta}$
with $\\{\sigma_{i}^{ab}\\}$ representing Pauli matrices. The Hamiltonian
contains the atomic and hopping parts ($\propto\epsilon_{a}$ and $t_{ij}$,
respectively), the so-called Hubbard term $\propto U$, representing the intra-
atomic interaction between the particles on the same atomic site _i_ with
opposite spins, the direct intersite Coulomb interaction $\propto K_{ij}$,
Heisenberg exchange $\propto J^{H}_{ij}$, and the two-particle and the
correlated hopping and intersite Coulomb terms ($\propto J_{ij}^{\prime}$ and
$V_{ij}$, respectively). The last term describes the ion-ion Coulomb
interaction which is adopted here in its classical form. The microscopic
parameters ($\epsilon_{a}$, $t_{12}\equiv t$, $U$, $K_{12}=K$,
$J^{H}_{12}={J}^{\prime}_{12}$ and $V_{12}=V$) are all calculated explicitly
in the resultant correlated state by readjusting the single-particle wave
function size contained in their expressions (for their analytic expressions
see Supporting Information). The primed summations are taken for $i\neq j$.
The evolution of the new introduced quantities: atomicity, true covalency, and
ionicity, is analyzed in detail as a function of interatomic distance.

The Hamiltonian (15) was determined by defining first the field operators
$\hat{\psi}_{\sigma}(\textbf{r})$ and
$\hat{\psi^{\dagger}}_{\sigma}(\textbf{r})$, i.e.,

$\displaystyle\begin{cases}\hat{\psi}_{\sigma}(\textbf{r})=\sum_{i\sigma}w_{i}(\textbf{r})\chi_{\sigma}(i)\hat{a}_{i\sigma},\\\
\hat{\psi^{\dagger}}_{\sigma}(\textbf{r})=\sum_{i\sigma}w_{i}(\textbf{r})\chi_{\sigma}^{\dagger}(i)\hat{a}^{\dagger}_{i\sigma},\end{cases}$
(16)

where $\hat{a}_{i\sigma}$ ($\hat{a}^{\dagger}_{i\sigma}$) are the annihilation
(creation) operators of the single-particle states
$w_{i}(\textbf{r})\chi_{\sigma}(i)$ on atom $i=1,2$ with spin $\sigma$. Note
that the single–particle basis is composed of the Hückel-Slater orbitals only;
this restriction represents the only approximation here; all the remaining
analysis and results are exact within these limitations. Those operators, in
turn lead to the expression (15) of $\hat{\mathcal{H}}$ in a standard manner.
To close the formal methodological part, the two-particle spin-singlet wave
function is defined as 14

$\displaystyle\psi_{G}(\textbf{r}_{1},\textbf{r}_{2})=\frac{1}{\sqrt{2}}\bra{0}\hat{\psi}_{1}(\textbf{r}_{1})\hat{\psi}_{2}(\textbf{r}_{2})\ket{\psi_{G}},$
(17)

where $\ket{0}$ is the universal vacuum state for particles and
$\ket{\psi_{G}}$ is the ground state, both in Fock space. This relation
provides equivalence of description both in terms of two-particle wave
function and the second-quantization language.

In Fig. 4 we provide the flowchart of the numerical part of our analysis which
concerns mainly the determination of the orbital size $\alpha^{-1}$ in the
interacting (correlated) state and, in effect, of the optimal ground state
energy and all other bonding–state characteristics.

This work was supported by Grants OPUS No. UMO–2018/29/B
/ST3/02646 and No. UMO–2021/41/B/ST3/04
070 from Narodowe Centrum Nauki. Discussions with our colleague from
Theoretical Chemistry Department, Prof. Ewa Brocławik and dr hab. Mariusz
Radoń from Inorganic Chemistry Department of the Jagiellonian University are
gratefully acknowledged. The authors are also grateful to Prof. Jurgen M.
Honig from Purdue University for his critical reading of the manuscript.

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The formal solution of Hamiltonian (15) is carried out explicitly by selecting
a trial basis in the Fock space for this two orbital systems. In the case with
$N=4$ spinorbitals and $N_{e}=2$ electrons we have
$\binom{N}{N_{e}}=\binom{4}{2}=6$ states in the occupation number
representation. They are

$\displaystyle\begin{cases}&\ket{1}=\hat{a}^{\dagger}_{1\uparrow}\hat{a}^{\dagger}_{2\uparrow}\ket{0},\\\
&\ket{2}=\hat{a}^{\dagger}_{1\downarrow}\hat{a}^{\dagger}_{2\downarrow}\ket{0},\\\
&\ket{3}=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{1\uparrow}\hat{a}^{\dagger}_{2\downarrow}+\hat{a}^{\dagger}_{1\downarrow}\hat{a}^{\dagger}_{2\uparrow})\ket{0},\\\
&\ket{4}=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{1\uparrow}\hat{a}^{\dagger}_{2\downarrow}-\hat{a}^{\dagger}_{1\downarrow}\hat{a}^{\dagger}_{2\uparrow})\ket{0},\\\
&\ket{5}=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{1\uparrow}\hat{a}^{\dagger}_{1\downarrow}+\hat{a}^{\dagger}_{2\downarrow}\hat{a}^{\dagger}_{2\uparrow})\ket{0},\\\
&\ket{6}=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{1\uparrow}\hat{a}^{\dagger}_{1\downarrow}-\hat{a}^{\dagger}_{2\downarrow}\hat{a}^{\dagger}_{2\uparrow})\ket{0}.\end{cases}$
(18)

The first three are the spin–triplet states, whereas the next three are the
spin–singlets and the mixture of those will result in the ground state. By
calculating the Hamiltonian matrix $\bra{i}\hat{\mathcal{H}}\ket{j}$ we obtain
a $6\times 6$ matrix which can be diagonalized analytically (see Ref. 4). In
the case of interest to us case we list only the ground state eigenvalue and
corresponding ground states which are

$\displaystyle\begin{cases}E\equiv\lambda_{5}=2\epsilon+\frac{1}{2}(K+U)+J-\frac{1}{2}D,\\\
\ket{\psi_{G}}=\frac{1}{[D(D-U+K)]^{\frac{1}{2}}}[(4(t+V))\ket{4}\\\
-(D-U+K)\ket{5}],\end{cases}$ (19)

where parameters are defined in the main text. Their explicit expressions are

(20)

where $\beta$ and $\gamma$ are mixing parameters and ${\epsilon}^{\prime}$,
${t}^{\prime}$, ${U}^{\prime}$, ${K}^{\prime}$, ${J}^{\prime}$, ${V}^{\prime}$
are analogous microscopic parameters calculating for single-particle wave
functions. They can be expressed as a function of interatomic distance $R$ and
inverse orbital size $\alpha_{0}$

(21)

with the Euler constant $C\simeq 0.57722$, and

$\displaystyle\begin{cases}E_{i}(x)=-\int^{\infty}_{x}\frac{dt}{t}\text{
exp}(-t),\\\ S=\text{exp}(-\alpha R)\left(1+\alpha
R+\frac{1}{3}\alpha^{2}R^{2}\right),\\\ S^{\prime}=\text{exp}(\alpha
R)\left(1-\alpha R+\frac{1}{3}\alpha^{2}R^{2}\right).\end{cases}$ (22)

In effect, the determination of $\ket{\psi_{G}}$ requires the readjustment of
the Slater-orbital size contained in the expression for Hückel-Wannier
orbitals $w_{i}(\textbf{r})$, a procedure of which is schematically
illustrated by the flowchart in Fig. 4.

The microscopic parameters obtained in this way are shown in Figs. 5 and 6.
Parenthetically, the mixing coefficients $\beta$ and $\gamma$ are expressed
through the overlap integral
$S\equiv\braket{\psi_{1}(\textbf{r})^{\dagger}\psi_{2}(\textbf{r})}$

$\displaystyle\begin{cases}\beta=\frac{1}{\sqrt{2}}\sqrt{\frac{1+\sqrt{1+S^{2}}}{1-S^{2}}},\\\
\gamma=\frac{S}{1+\sqrt{1-S^{2}}}.\\\ \end{cases}$ (23)

Figure 5: Microscopic parameters indicated versus interatomic distance $R$.
The equilibrium bond length and the onset of Mottness, $R_{Mott}$ are also
marked.

The whole procedure is closed once we calculate explicitly the ground-state
two-particle wave function according to the general rule

$\displaystyle\psi_{G}(\textbf{r}_{1},\textbf{r}_{2})=\frac{1}{2}\bra{0}\\{\hat{\psi}_{\uparrow}(\textbf{r}_{1})\hat{\psi}_{\downarrow}(\textbf{r}_{2})$
$\displaystyle-\hat{\psi}_{\downarrow}(\textbf{r}_{1})\hat{\psi}_{\uparrow}(\textbf{r}_{2})\\}\ket{\psi_{G}}$
(24)

Figure 6: Microscopic parameters, $J^{H}$ and $V$ versus $R$ (same
characteristics as in Fig.(5)).

The final result is the expression (1), which contains single-particle
molecular wave functions $\\{w_{i}(\textbf{r})\\}_{i=1,2}$.

The problem one encounters at the outset is that if one examines the $R$
dependence of the coefficients $|C|^{2}/(|C|^{2}+|I|^{2})$ and
$|I|^{2}/(|C|^{2}+|I|^{2})$; this has been plotted in Fig. 7. Namely, one
spots that such naturally defined covalency exhibits a clear unphysical
behavior in the limit $R\rightarrow\infty$, when the molecular states should
reduce to the parent atomic Slater states. To avoid this basic deficiency the
concept of atomicity was involved and discussed in detail in the main text.

Figure 7: Two-particle covalency and ionicity for H2 molecule calculated as
$|C|^{2}/(|C|^{2}+|I|^{2})$ and $|I|^{2}/(|C|^{2}+|I|^{2})$, respectively.
These results are reinterpreted subsequently and the results, including
atomicity, are displayed in Fig. 2 in the main text.

Along with the explicit wave function expression (cf. Eqs.(1) and (5)) in main
text), one can obtain the corresponding expression for the physical densities
(12)–(14).