Mole ular orbitals and strong-(cid:28)eld approximation Thomas Kim Kjeldsen and Lars Bojer Madsen Department of Physi s and Astronomy, University of Aarhus, 8000 Århus C, Denmark V.I.Usa henkoandS.-I.Chu[Phys. Rev. A71,063410(2005)℄dis ussthemole ularstrong-(cid:28)eld approximationinthevelo itygaugeformulationandindi atethatsomeofourearliervelo itygauge 6 al ulationsareina urate. Herewe ommentontheresultsofUsa henkoandChu. First,weshow 0 that the mole ular orbitals used by Usa henko and Chu do not have the orre t symmetry, and 0 se ond, that it is an oversimpli(cid:28) ation to des ribe the mole ular orbitals in terms of just a single 2 linear ombination of two atomi orbitals. Finally, some values for the generalized Bessel fun tion are givenfor omparison. n a PACSnumbers: 33.80.Rv,32.80.Rm J 3 2 I. INTRODUCTION II. MOLECULAR ORBITALS ] h The idea of the LCAO approa h to mole ular orbital p theory is to identify the mole ular orbitals in a basis of - The mole ular strong-(cid:28)eld approximation (MO-SFA) m wasformulatedinthevelo itygaugeinRef.[1℄,andmore atomi orbitals. For a homonu learΨdi(art)omi mole ule, we write a mole ular wave fun tion as a superpo- o re ently in the length gauge [2℄. The latter work shows sition of atomi orbitals whi h are entered on ea h of t that the velo ity gauge MO-SFAwith the initial stateof a the atomi ores. The nu lei arelo ated at the positions . thehighesto upiedmole ularorbital(HOMO)obtained ±R/2 s with respe t to the enter of the mole ule. The in a self- onsistent Hartree-Fo k (HF) al ulations does c mole ularorbitals arethe eigenfun tions of the Fo k op- i not a ount for the observed orientation-dependent ion- s 2 erator, i.e., of the kineti energy operator and the full y ization of N [3℄. Only the orresponding length gauge HFpotential. Theseeigenfun tions anbeobtainedbya h MO-SFA [2℄ and the mole ular tunneling theory [4℄ give diagonalization of the matrix representation of the Fo k p the orre t behavior. φnlm(r ∓R/2) [ operator in the atomi entered basis, . A great simpli(cid:28) ation of the problem is a hieved if the 2 The orientation dependen e of the strong-(cid:28)eld ioniza- basis fun tions are symmetrized a ording to the mole - 3v tisiomnaodfeN2fowratshreev3iσsgiteHdOinMROefi.n[5t℄.erTmhseroefaassiminpglleemlinoedaerl [uφla10r0(proi∓ntRg/ro2u)℄po(rDbi∞tahl)s.. CAon hsiadnegre, eo.fg.b,atshise laetaodmsit o1as 1 2 p ombinationofatomi orbitals(LCAO)usingtwoatomi new set of basis fun tions 8 orbitals. Then the MO-SFA is applied in the velo ity R R 0 gaugeto al ulatetheionizationrate,anditis on luded ψσg1s(r) = Nσg1s[φ100(r− 2 )+φ100(r+ 2 )] (1) 5 that the velo ity MO-SFA, with su h an approximation R R 0 fortheHOMO,is apableofpredi tingtheobservedmin- ψσu1s(r) = Nσu1s[φ100(r− 2 )−φ100(r+ 2 )]. (2) s/ imum [3℄ in the rate for perpendi ular orientation of the c internu learaxiswithrespe ttothelinearpolarizationof si thelaser. Basedonthese(cid:28)ndingsitisstated[5℄thatour These new basisDfu∞nh tions beloΣn+gg to de(cid:28)nite symmetry y earlier velo ity gauge al ulations [2℄ are ina urate be- Σla+ssi(cid:28) ationsin ,namely ('+' ombination)and u h auseof ina ura iesin our(i) mole ularorbitaland (ii) ('-' ombination). The 2s, 3s, et . orbitals are sym- p metrized similarly. If we extend the basis to ontain generalized Bessel fun tion. Here we refute both these 2p : atomi orbitals,the properlysymmetrizedbasisfun - v laims by giving details of our orbitals and generalized Σ i Bessel fun tions. Additionally, we show that apart from tions in the blo ks are X 3σg R R r having the w2rong symmetry, the model used for the ψσg2p(r) = Nσg2p[φ210(r− 2 )−φ210(r+ 2 )] (3) a HOMO of N in Ref. [5℄ is too simple and we therefore R R biedleienvtealt.haTtothae aogurnetemfoerntthine [m5℄olwe ituhlaerxpsterruim teunrteims oar e- ψσu2p(r) = Nσu2p[φ210(r− 2 )+φ210(r+ 2 )],(4) LCAOs have to be in luded as, e.g., in HF al ulations. 2p0 2pz With su h a self- onsistent wave fun tion, the velo ity forthe ( ) orbitals,while the `+'and `(cid:21)' ombina- 2p±1 πu πg gaugeMO-SFA,indeedgivesthewrongpredi tionforthe tionofthe areof and symmetry,respe tively, orientation-dependent ionization [2℄. Finally, we present and hen e not of interest in our present dis ussion fo- 3σg 2 a urate values from [2℄ of the generalized Bessel fun - ussing on the HOMO of N . tion sin e also in this ase some dis repan ies exist with ItisimportanttonotethatforaquantitativeHF al u- Ref. [5℄ (cid:21) a dis repan y whi h an be attributed to the lation, the basis is further extended until onvergen e is g u di(cid:27)eren e in wavelength (we use 800 nm light [2℄, in [5℄ obtained. Wemayeasily he kthe orre tparity( or ) r →−r 795 nm light is used [6℄). ofthebasisfun tionbyletting andusingthefa t 2 basis fun tions themselves to be eigenfun tions, instead −3 −3 −2 (a) 0.4 −2 (d) 0.4 the eigenfun tions are linear ombinations of basis fun - −1 0.2 −1 0.2 tions. x 0 0 qx 0 0 12 −0.2 12 −0.2 −0.4 3 −0.4 3 −3−2−1 0 1 2 3 −3−2−1 0 1 2 3 q z z −3 −3 (b) 0.4 (e) 0.4 III. HARTREE-FOCK CALCULATIONS −2 −2 −1 0.2 −1 0.2 x 0 0 qx 0 0 1 −0.2 1 −0.2 In Ref. [2℄ we used the GAME2SS [9℄ pr(o1g1rsa6mp)f/o[5rso3upr] 2 2 quantitativeHF al ulationsofN witha 3 −0.4 3 −0.4 Cartesian Gaussian basis [10℄. The way we spe ify −3−2−1 0 1 2 3 −3−2−1 0 1 2 3 z qz our basis is standard in the quantum hemistry liter- −3 (c) 0.4 −3 (f) 0.4 ature [11℄, but is re alled here fsor omppleteness. The −2 −2 wave fun tion is expanded in 11 and 6 orbitals en- −1 0.2 −1 0.2 tered on ea h atom. These basis orbitals all have dif- x 0 0 qx 0 0 ferent exponentials in the radial parts. The orbitals of 1 −0.2 1 −0.2 ourHF al ulationareobtainedby avariationoftheex- 2 2 −0.4 −0.4 pansion oe(cid:30) ients until the energy is minimized. This 3 3 −3−2−1 0 1 2 3 −3−2−1 0 1 2 3 pro edureis equivalent to an iterativediagonalizationof z qz matrix representations of the Fo k operator. With lit- tlelossofa ura ybut signi(cid:28) antgainin omputational speed, theoptimizationpro edure anbesimpli(cid:28)edwith FIG. 1: (Color online). Planar uts through the 3σg HOMO a ontra ted basis set. This means that all the expan- 2 (12s7p)/[6s4p] of N obtainedin a HF al ulation with a ba- sion oe(cid:30) ients are not varied independently, ontrary sis (see text). Left (right) olumn: oordinate (momentum) a (cid:28)xed relationship is kept between some of the oe(cid:30)- s spa e wave fun tion. Upper row: ontribution pfrom type ients. Su h a simpli(cid:28) ation is used in most quantums basis fun tions, middle row: ontribution from(12st7ypp)/e[6bsa4spis] hemistry software. Five oe(cid:30) ients orrespondingpto fun tions, lower row: total HF result in the orbitals and three oe(cid:30) ients orresponding to the or- basis. bitals are optimized independently. In order to des ribe theasymptoti behaviorweaugmentthebasisbyadding φ (−1)l s p nlm an extra di(cid:27)use and orbital. In total, the wave fun - thattheparityof is . Weemphasizethispoint s p 2p z tion is e(cid:27)e tively expanded in a basis of six and four sin e in Ref. [5℄, the `+' ombination of the orbitals (12s7p)/[6s4p] orbitals on ea h atom denoted by basis. is erroneously referred to as the gerade state. Although not of relevan e for our fully numeri al HF al ulations, In Fig. 1 we show results of our al ulations for the we note that in the hydrogeni basis used in Ref. [5℄ the 3σg 2 HOMO of N [Fig. 1 ( )℄ and in its de omposition s p Fourier transform of the antisymmetri ombination of 2p of [Fig. 1 (a)℄ and [Fig. 1 (b)℄ type basis fun tions. z atomi orbitals is Weseethattheorbitalis omposedofbothtypesofbasis p p ψ˜σg2p(p)=Nσg2psin(p·R/2)(1+4p2)3Y10(pˆ), (5) fun tions. At(cid:28)rstsightthe ontributionaloneseemsto resemble the orbital quite well. However, in the proxim- s ity of the nu lei, it is the ontribution that dominates, with Nσg2p the normalization onstant. We may ver- sin ethepfun tions havenodesonthenu lei. Itisthes p → −p ify theYger(a−dpˆe)p=ar(it−y1s)ylYmm(epˆt)ry by letting and type ontribution that dominates the momentum spa e lm lm using . Equation (5) is the or- wave fun tion at low momenta. This fa t be omes lear re tly symmetrized version of Eq. (20) of Ref. [5℄. The in Figs. 1 (d)-(f), where we show the Fourier transforms di(cid:27)eren eliesinthesine-fa torofEq.(5) omparedwith ofthefun tionsinFigs.1(a)-( ). Itis learthatboththe p s p a osine-fa tor in Ref. [5℄. A al ulation with the -type and typeofbasisfun tions ontributesigni(cid:28) antlyto basis fun tions alone would, be ause of this sine-fa tor, the momentum wave fun tion. Therefore, as dis ussed 2 predi t suppressed ionization of N ompared with the in Se . II, it is insu(cid:30) ient to onsider ea h type ex lu- s p z ompanion Ar atom (cid:21) in ontrast to experimental and sively: The and basis orbitals are not separately theoreti al (cid:28)ndings [1, 7, 8℄. Thus having pointed out eigenfun tions of the Fo k operator, ontrary, the mix- the importan e of working with properly symmetrized ture turns out to be essential. At this point the authors basis fun tion, we now onfront the ad ho single LCAO of Ref. [5℄ introdu e an oversimpli(cid:28) ationby onsidering appro h [5℄ with our HF al ulations. Clearly, but on- the HOMO to be onstru ted only from a single linear p trary to the approa h in Ref. [5℄, one annot expe t the ombination of two -orbitals. 3 IV. IMPLICATIONS TO THE MOLECULAR 1.4 Grid basis (a) s+p STRONG-FIELD APPROXIMATION 1.2 s only 0) p only θe( = 1.0 IntheMO-SFAone al ulatesthetransitionratefrom at 0.8 R theHOMnOtoa ontinuumVolkovstatethroughabsorp- θe() / 0.6 tion of photons. For aωlinearly polarized,Fm0ono hro- Rat 0.4 mati (cid:28)eld of frequen y and (cid:28)eld strength , the ve- 0.2 lo ity gauge result for the angular di(cid:27)erential ionization 0.0 0 30 60 90 120 150 180 rate in atomi units is [1℄ θ (degrees) dW 2 = 2πN p Ψ˜(p ) (U −nω)2 1.4 Grid basis dpˆ e n(cid:12) n (cid:12) p 1.2 (b) s osn+lpy (cid:12) (cid:12) 0) p only × Jn2(cid:18)|F0ω(cid:12)·2pn|,−(cid:12)2Uωp(cid:19), (6) θate( = 01..80 R wpit=h Ne2t(hneωn−umUbe−rIof)ele trons o upyinUg t=heFH2/O(4MωO2), θRate() / 00..46 n p p p 0 0.2 the momentum, p Ip 0.0 the ponderom0o.5t7iv2e5energy, the io2nizationΨ˜(ppo)tential of 0 30 60 θ (de 9g0rees) 120 150 180 the HOMO ( in the aseofN [12℄), the mo- J (u,v) n mentum spa e wave fun tion and a generalized Bessel fun tion [13℄. 2×1014 2 2 A ordingto the dis ussion in Se . III, it is important F3IσGg.2: Ionizationrateat θ W m oftheHOMOofN ( )asafun tionoftheangle betweentheinternu learaxis to apply the orre t momentum spa e wave fun tion in andthepolarizationaxisat(a)800nm[2℄and(b)795nm[6℄. the evaluation of Eq. (6). To illustrate this fa t, we on- Theshort-dashedandsolid urvearethe al ulationswiththe sider the total ionization rate integrated over all angles HF wave fun tion derived from a grid al ulation and with of the outgoing ele tron and summed over all numbers (12s7p)/[6s4p] the basis set in Fig. 1 (f), respe tively. The of a essible photon absorptions. We present this to- θ long-dashed urvewsithamaximumpandthe hained urveare tal ionization rate for di(cid:27)erent angles between the in- obtainedwiththe [Fig.1(d)℄and [Fig. 1(e)℄ ontributions (θ=0) t3eσrgnu lear axis a2nd the polarization axis for the HOMO only,respe tively. Allratesarenormalizedtoparallel orbital of N . We onsider three des riptions of the geometry. s mole ular orbital: (i) using -type basis fun tions only, p (ii) using -type basis fun tions only and (iii) using the self- onsistent HF solution. Clearly the approa hes (i) horizontalin Fig.1(e)andoverlapswithalargevalueof and (ii) only give a very poor des ription of the HOMO themomentumwavefun tion. Finally,weturntothefull s and therefore these results deviate from the result ob- wavefun tion in ase(iii) whi histhe oherentsum of p tained by using the true HOMO. In Fig. 2 we present and ontributions,Fig.1(f). Itisthelowmomentathat the results for the three ases at (a) 800 nm [2℄ and (b) ontribute mostly to the total rate. At these small mo- 795nm[6℄. Additionally,weshowinFig.2theresultsob- menta,weseefrom Fig. 1(f) thatthe absolutesquareof tained with an initial HOMO derived from a grid-based themomentumwavefun tionismaximizedalongthedi- HF al ulation [14℄. The ex ellent agreement with the re tionperpendi ulartothemole ularaxisandtherefore (12s7p)/[6s4p] basis-state al ulation proves the onver- the rate is maximized with the mole ule aligned perpen- θ = π/2 gen eofthelatterapproa h. Wenotethatpanels(a)and di ularly to the laser polarization, . This result (b)areverysimilarandhen ethedi(cid:27)eren esbetweenthe is also shown in Fig. 2 verifying the result of Ref. [2℄. results in [2℄ and [5℄ are not due to the slight di(cid:27)eren e Note that the rate using the total wave fun tion is not s p in wavelength. the sum of rates of the and ontributions sin e their FromFig.1(d)weseethatthe momentumwavefun - ontributions to the wave fun tion should be added o- tionin ase(i)isnearlyspheri allysymmetri ,andhen e, herently and witΨ˜h(pth)e orre t self- onsistent amplitudes in Fig. 2 the ionization rate in this ase is nearly inde- before entering in Eq. (6). We note that the ad- pendent of the mole ular orientation. In ase (ii) we use mixture of atomi s and p orbitals for the des ription of 2 the momentum fun tion from Fig. 1 (e). Although ex- theHOMOofN wasalsore entlypointedoutinrelated eptions o ur,asevident from Fig. 3(a) below, the gen- work on high-harmoni generation [15℄. p eralizedBesselfun tionsaretypi allymaximizedwhen In Refs. [2, 5℄, the values of the generalized Bessel isnearlyparalleltothepolarizationdire tion[Fig.3(b)℄. fun tion in Eq. (6) do also not agree. Figure 3(a) In a qualitative analysis, one may therefore expe t that shows our previous result for the square of the general- themaximumrateisobtained,wh|eΨ˜n(tph)e|2polarizationaxis ized Bessel fun tion orresponding to 128×ph1o0t1o4n absorp2- oin ides with a dire tion where is large. Corre- tion at 800 nm and at an intensity of W/ m . spondingly, in ase (ii) we see from Fig. 2 that the rate Panel (b) shows the orresponding result at 795 nm [6℄. θ = 0 is maximized when , i.e. when the polarization is Hen e,Panel(b) orrespondstoFig.2(a)ofRef.[5℄(sin e 4 6 x 10-7 generalized Bessel fun tion are orre t. Due to the ex- (a) ellentagreementat 795nm with the Besselfun tionsof 4 x 10-7 Ref. [5℄ we likewise believe that the latter are evaluated orre tly. 2 x 10-7 0 x 100 2 x 10-7 V. CONCLUSION 4 x 10-7 In on lusion,thevelo itygaugeMO-SFAwhen based on atransitionto the ontinuumfrom an initial HOMO, 6 x 10-7 determinedbyaHF al ulationusingstandardquantum 6 x 10-7 hemistry software, gives the wrong predi tion for the (b) orientation-dependent rate. In this approximation only 4 x 10-7 the length gauge MO-SFA (and the mole ular tunneling theory)[2℄ givethe observed minimum [3℄ in the rate for 2 x 10-7 the perpendi ular geometry. This on lusion ontrasts the (cid:28)ndings in a re ent work [5℄. In that work, however, 0 x 100 p only a single LCAO of two -orbitals (of wrong symme- 2 x 10-7 try)was onsideredforthedes riptionoftheHOMOand noself- onsistentwavefun tionwasapplied. FromFig.2 4 x 10-7 we learly see, that an appli ation of a too simple wave p fun tion, e.g., one with only basis fun tions, leads to 6 x 10-7 a predi tion a identally in qualitative agreement with experiment [3℄. If one nevertheless would be en ouraged p by this agreement to believe that the single LCAO of orbitals des ribes the physi s well, one should note that FnIG=.138: Polarplotof2th×eg1e0n1e4ralized B2essel fun tionoforder su h a wave fun tion due to the presen e of the sine- squared for W/ m , and (a) 800 nm [2℄, (b) 795 nm [6℄. The polar angle θp is measured from the fa tor in (5) would predi t suppressed ionization in N2 horizontal line. omparedwith Ar whi h is in ontrastwith experiments and theory [1, 7, 8℄. In losing we note that also in the J−n(u,v)=(−1)nJn(u,−v) atomi ase,the lengthgauge versionof the SFA wasre- [13℄) and by omparison we entlyshowntobesuperiortothevelo itygaugeversion see that the results agree. We may therefore attribute when ompared with results obtained by integrating the the di(cid:27)eren e between the generalized Bessel fun tions time-dependent S hrödinger equation [17℄. in [2℄ and [5℄ to stem from the di(cid:27)eren e in the wave- lengths used. It is lear from Fig. 3 that the values of the generalized Bessel fun tion are very sensitive to the A knowledgments exa t value of the arguments. We reprodu e the values forthegeneralizedBesselfun tiongiveninRef.[16℄with the input parameters used therein, and we therefore be- LBM is supported by the Danish Natural S ien e Re- lieve that our previous and present al ulations of the sear h Coun il (Grant No. 21-03-0163). [1℄ J. Muth-Böhm, A. Be ker, and F. H. M. Faisal, Phys. At. Mol. Opt. Phys.29, L677 (1996). Rev. 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