The van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
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1 Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor dpth. It provids a nic application of prturbation thory. Two atoms W considr two atoms and. tom has a cntr of mass position R and constitunts at positions r a + R. That is, r a is th diffrnc btwn th constitunt position and th cntr of mass position. Th constitunts hav chargs a and masss m a. Th indics a ar in a st that w will call. On of th constitunts is th nuclus. Th rlativ position r a for th nuclus is vry small (th nuclus is clos to th cntr of mass), so w can tak r a for th nuclus to b zro. If th atom is nutral, thn a = 0. W will b most intrstd in that cas, but w will also considr th possibility of an ion, for which th total charg is not zro. tom has a cntr of mass position R and constitunts at positions r b + R. Th constitunts hav chargs b and masss m b. Th indics b ar in a st that w will call. Th hamiltonian is H = H + H + V whr H is th hamiltonian for atom by itslf, H = p a 2 + 2m a 2 H is th hamiltonian for atom by itslf, H = a Copyright, 202, D. E. Sopr 2 sopr@uorgon.du p b 2 + 2m b 2 a a a b b b a a r a r a b b r b r b, (), (2)
2 and V is th intraction btwn th two atoms, V = a b R n + r a r b, (3) whr R = R a R b (4) and n is a unit vctor in th dirction of R a R b R n = R a R b. (5) W would lik to analyz this situation in th orn-opnhimr approximation. Th atoms xrt a forc on ach othr. W think of th atoms as moving slowly and th lctrons in ach atom as moving quickly. Thn as th atoms mov, th lctron orbits in ach atom adjust vry quickly to th sparation R n btwn th atoms. Th atom nrgy lvls thus slowly chang. This shift in nrgy thn constituts an ffctiv potntial V O (R) btwn th atoms. This ffctiv potntial is, approximatly, a rplacmnt for th xact V. Th orn-opnhimr approximation is prtty intuitiv, but w will not xplor it furthr in ths nots. Rathr, our task hr is to dtrmin how V O (R) dpnds on th sparation R. W assum that both atoms ar in thir ground stats and that th sparation R is larg compard to th sizs of th atoms. For this purpos, w tak th sparation R n to b fixd. 2 Th intraction for larg sparation Having st up th problm, w now nd to work out what V looks lik whn th sparation R n is larg compard to th intra-atomic sparations r a and r b. That is, w ask how Φ(R n + ) = R n + = R [ + 2 n /R + 2 /R 2 ] /2 (6) bhavs whn R. Using + ɛ = 2 ɛ ɛ2 +, (7) 2
3 w hav Φ(R n + ) = R [ 2 (2 n /R + 2 /R 2 ) + 3 ] 8 (2 n /R) 2 +. (8) That is, Φ(R n + ) = R This has th form whr Φ(R n + ) = R [ R n [ 2 3( n 2R ) ] ] [ R P ( n) i i ] 2R P 2( n) ij i j + 2 P ( n) i = n i, P 2 ( n) ij = δ ij 3n i n j.. (9), (0) () This sris gos on. Th nxt trm has th form const. P 3 ( n) ijk i j k /R 3. W not that P ij 2 is symmtric undr intrchang of i, j and is traclss: P ij 2 δ ij = 0. Th highr ordr tnsors P J ar also compltly symmtric and traclss in all pairs of indics. s w will s, this xpansion is rlatd to multipol momnts. W can us this to xpand our intraction V, using Q = a a and Q = b b, V = a b R n + r a r b = a b R R 2 a b P ( n) i (ra i rb) i 2R 3 a b P 2 ( n) ij (ra i rb)(r i a j r j b ) + = Q Q R + Q R 2 b P ( n) i rb i Q R 2 a P ( n) i ra i Q 2R 3 b P 2 ( n) ij rbr i j b Q 2R 3 a P 2 ( n) ij rar i a j + R 3 a b P 2 ( n) ij rar i j b +. 3 (2)
4 3 First ordr prturbation thory W can valuat th intraction nrgy in first ordr prturbation thory by simply taking th xpctation valu V of V in th ground stats of th two atoms. Th biggst trm is Q Q /R. This is th lctrostatic attraction or rpulsion btwn th two atoms. If Q and Q ar nonzro, this trm is largr than any of th othr trms (in an xpansion in powrs of (atom siz)/r). W will mostly b intrstd in two nutral atoms, Q = Q = 0. Howvr, first, it is intrsting to considr th cas that Q is zro but Q is not. Thn what contribution is biggst for larg R? Thn thr is a /R 2 trm proportional to P ( n) i a r i a. (3) This trm dscribs th intraction of th lctric dipol momnt of atom with th fild from atom. Howvr th xpctation valu in th ground stat of th lctric dipol momnt of atom must vanish. If th total angular momntum of atom is zro, thn this statmnt follows from rotational invarianc. Evn if th total angular momntum of atom is not zro, parity invarianc prvnts th dipol momnt from bing non-zro as long as th ground stat is a parity ignstat. Furthrmor, th ground stat must b a parity ignstat as long as it is not dgnrat. Thus w can safly rul out a non-vanishing /R 2 trm. Thr is a /R 3 trm proportional to P 2 ( n) ij a r j ar j a. (4) This trm dscribs th intraction of th lctric quadrupol momnt of atom with th fild from atom. If atom has angular momntum zro, thn rotational invarianc givs a rar j a j δ ij. (5) ut P 2 ( n) ij δ ij = 0, so w gt a contribution zro. 3 If atom has, say, angular momntum, thn I think that it can hav a quadrupol momnt and w can gt a non-zro contribution. 3 ctually, w also gt a zro contribution if th total angular momntum is /2, but th 4
5 Thr is anothr /R 3 contribution, proportional to P 2 ( n) ij a ra j b r j b. (6) This trm is prsnt vn if th total charg of both atoms vanish. This contribution vanishs bcaus th xpctation valus of th dipol momnts vanish. t th ordr /R 4, w gt contributions lik P 3 ( n) ijk a rar j a j b rb k. (7) t th ordr /R 5, w gt a trm proportional to P 4 ( n) ijkl a rar j a j b rb k rb l. (8) Trms lik this ar thr vn if th total charg of both atoms vanish. Th dipol momnt factor b rb k (9) will vanish by th parity argumnt givn prviously. For th /R 5 trm in Eq. (8), w could possibly hav a non-zro contribution. Howvr, if th angular momnta of th ground stats vanish, thn Eq. (5) applis and th contribution vanishs bcaus P 4 ( n) ijkl δ kl = 0. In fact, w s that thr is a rmarkabl consqunc if th ground stats hav angular momntum zro: all of th contributions to V vanish. This mans that w should look at scond ordr prturbation thory. 4 Scond ordr prturbation thory W hav sn that if th atoms hav zro charg, ar in thir ground stat, and hav angular momntum zro, V = 0 and w nd to go to scond argumnt is a littl mor subtl. a P 2 ( n) ij r j ar j a is an angular momntum 2 oprator. If w combin angular momntum 2 with angular momntum /2 for th kt stat, w can angular momntum 3/2 or 5/2, but w cannot gt angular momntum /2 to match th bra stat. 5
6 ordr prturbation thory to look for th nrgy shift. With Q = Q = 0, th intraction V is V = R 3 a b P 2 ( n) ij rar i j b +. (20) Th corrsponding scond ordr nrgy shift is = R 6 a,a b,b a a b b L 0 K 0 P 2 ( n) ij P 2 ( n) kl 0, r l a K, K, r j a 0, 0, r k b L, L, r i b 0, E () 0 + E () 0 E () K E() L. (2) Hr 0, is th ground stat for atom and th stats K, ar th stats othr than th ground stat. n analogous notation applis for atom. Th sums ovr stats of th atoms, as dictatd by th projction oprator Q, includs all stats {K, L} xcpt th stat {0, 0} consisting of both atoms in thir ground stats. In our application, nithr {0, L} for L 0 nor {K, 0} for K 0 contribut bcaus th dipol momnt oprator has vanishing matrix lmnt btwn 0 and 0. W can think of this nrgy shift as rprsnting a dipol-dipol intraction, whr th two atoms induc dipol momnts in ach othr. It is not so asy to find approximations for th matrix lmnts and sums hr. Howvr, vn without doing that, w larn somthing intrsting: nutral atoms will xprinc a /R 6 potntial at larg sparations. This is known as th van dr Waals potntial. Notic somthing ls. W gt a /R 6 potntial. If w usd trms in V with mor powrs of /R or if w usd highr ordr in prturbation thory, w would gt additional contributions, which would b proportional to /R n with n > 6. This tlls us that if w xpand V O (R) in powrs of (atom siz)/r, th first contribution is proportional to /R 6 and that th rsult in Eq. (2) givs th /R 6 contribution xactly. 6
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