Asymptotics of the ground state energy of heavy molecules and related topics. II

1.1 MB·English

by  Victor Ivrii

Asymptotics of the ground state energy of
heavy molecules and related topics
2
1
0
2 Victor Ivrii

t
c
O October 5, 2012

4

]
h Abstract
p
-
h We consider asymptotics of the ground state energy of heavy
t
a atoms and molecules in the strong external magnetic field and derive
m
it including Schwinger and Dirac corrections (if magnetic field is
[ not too strong). In the next instalment we extend this paper to

1 consider also related topics: an excessive negative charge, ionization
v
energy and excessive negative charge when atoms can still bind into
9
molecules.
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3
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.
0
Contents
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2
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:
v
Contents 1
i
X
r 0 Introduction 3
a
0.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2 Problems to consider . . . . . . . . . . . . . . . . . . . . . . 4
0.3 Magnetic Thomas-Fermi theory . . . . . . . . . . . . . . . . 5
0.4 Main results sketched and plan of the chapter . . . . . . . . 7
1 Magnetic Thomas-Fermi theory 8
1.1 Framework and existence . . . . . . . . . . . . . . . . . . . . 8
1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1
2 Applying semiclassical methods: M = 𝟣 15
2.1 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Smooth approximation . . . . . . . . . . . . . . . . . . . . . 22
2.3 Rough approximation . . . . . . . . . . . . . . . . . . . . . . 27
3 Applying semiclassical methods: M ≥ 𝟤 41
3.1 Analysis in zone 𝒳 . . . . . . . . . . . . . . . . . . . . . . . 41
𝟤
4 Analysis in the boundary strip 59
4.1 Properties of W𝖳𝖥 as N = Z . . . . . . . . . . . . . . . . . . 59
B
4.2 Semiclassical analysis in the strip 𝒴 for N ≥ Z. . . . . . . . 64
4.3 Analysis in the boundary strip 𝒴. Construction of the poten-
tial for N < Z . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Ground state energy 73
5.1 Lower estimates . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Upper estimate: general scheme . . . . . . . . . . . . . . . . 75
5.3 Upper estimate as M = 𝟣 . . . . . . . . . . . . . . . . . . . . 75
5.4 Upper estimate as M ≥ 𝟤 . . . . . . . . . . . . . . . . . . . . 78
6 Negatively charged systems (coming) 89
7 Positively charged systems (coming) 89
8 Appendices 89
8.A Electrostatic inequalities . . . . . . . . . . . . . . . . . . . . 89
8.B Very strong magnetic field case . . . . . . . . . . . . . . . . 94
8.C Riemann sums and integrals . . . . . . . . . . . . . . . . . . 95
2
Bibliography 95
0 Introduction
In this paper we repeat analysis of the previous paper [Ivr10]1) but in the
case of the constant external magnetic field2).
0.1 Framework
Let us consider the following operator (quantum Hamiltonian)
∑︁ ∑︁
(0.1.1) 𝖧 = 𝖧 := H + |x −x |−𝟣
N A,V,xj j k
𝟣≤j≤N 𝟣≤j<k≤N
on
⋀︁
(0.1.2) H = H , H = L𝟤(ℝd,ℂq)
𝟣≤n≤N
with
(︀ )︀𝟤
(0.1.3) H = (i∇−A)·𝝈 −V(x)
V,A
describingN sametypeparticlesintheexternalfieldwiththescalarpotential
−V and vector potential A(x), and repulsing one another according to the
Coulomb law.
Here x ∈ ℝd and (x ,...,x ) ∈ ℝdN, potentials V(x) and A(x) are
j 𝟣 N
assumed to be real-valued. Except when specifically mentioned we assume
that
∑︁ Z
m
(0.1.4) V(x) =
|x −𝗒 |
m
𝟣≤k≤M
where Z > 𝟢 and 𝗒 are charges and locations of nuclei. Here σ =
k k
(σ ,σ ,σ ), σ are q ×q-Pauli matrices.
𝟣 𝟤 d k
So far in comparison with the previous paper [Ivr10] we only changed
(24.1.3) of [Ivr11] to (0.1.3) introducing magnetic field. Now spin enters not
only in the definition of the space but also into operator through matrices
1) Coinciding with Chapter 24 of of [Ivr11].
2) Actually we need a magnetic field either sufficiently weak or close to a constant on
the very small scale.
3
σ . Since we need d = 𝟥 Pauli matrices it is sufficient to consider q = 𝟤 but
k
we will consider more general case as well (but q should be even).
Remark 0.1.1. In the case of the the constant magnetic field ∇×A
(︀ )︀𝟤
(0.1.5) H = −i∇−A(x) +𝝈 ·∇×A−V(x)
A,V
In the case d = 𝟤 this operator downgrades to
(︀ )︀𝟤
(0.1.6) H = −i∇−A(x) +𝜎 (∇×A)−V(x)
A,V 𝟥
Again, let us assume that
(0.1.7) Operator 𝖧 is self-adjoint on H.
As usual we will never discuss this assumption.
0.2 Problems to consider
As in the previous Chapter we are interested in the ground state energy
𝖤 = 𝖤 of our system i.e. in the lowest eigenvalue of the operator 𝖧 = 𝖧
N N
on H:
(0.2.1) 𝖤 := 𝗂𝗇𝖿𝖲𝗉𝖾𝖼𝖧 on H;
more precisely we are interested in the asymptotics of 𝖤 = 𝖤(𝗒;Z;N) as V
N
is defined by (0.1.4) and N ≍ Z := Z +Z +...+Z → ∞ and we are going
𝟣 𝟤 M
to prove that3) 𝖤 is equal to Magnetic Thomas-Fermi energy ℰ𝖳𝖥, possibly
B
with Scott and Dirac-Schwinger corrections and with appropriate error.
We are also interested in the asymptotics for the ionization energy
(0.2.2) 𝖨 := −𝖤 +𝖤
N N N−𝟣
and we also would like to estimate maximal excessive negative charge
(0.2.3) 𝗆𝖺𝗑 N −Z.
N: 𝖨 >𝟢
N
3) Under reasonable assumption to the minimal distance between nuclei.
4
All these questions so far were considered in the framework of the fixed
positions 𝗒 ,...,𝗒 but we can also consider
𝟣 M
(0.2.4) 𝖤̂︀ = 𝖤̂︀ = 𝖤̂︀(𝗒;Z;N) = 𝖤+U(𝗒;Z)
N
with
∑︁ ZmZm′
(0.2.5)
|𝗒 −𝗒 |
m m′
𝟣≤m<m′≤M
and
(0.2.6) 𝖤̂︀(Z;N) = 𝗂𝗇𝖿 𝖤̂︀(𝗒;Z;N)
𝗒𝟣,...,𝗒M
and replace 𝖨 by ̂︀I = −𝖤̂︀ +𝖤̂︀ and modify all our questions accord-
N N N N−𝟣
ingly. We call these frameworks fixed nuclei model and free nuclei model
respectively.
In the free nuclei model we can consider two other problems:
- Estimate from below minimal distance between nuclei i.e.
𝗆𝗂𝗇 |𝗒 −𝗒 |
m m′
𝟣≤m<m′≤M
for which such minimum is achieved;
- Estimate maximal excessive positive charge
∑︁
(0.2.7) 𝗆𝖺𝗑{Z −N : 𝖤̂︀ < 𝗆𝗂𝗇 𝖤(Z ;N )}
m m
N
N𝟣,...,NM: 𝟣≤m≤M
N𝟣+...NM=N
for which molecule does not disintegrates into atoms.
0.3 Magnetic Thomas-Fermi theory
As in the previous Chapter the first approximation is the Hartree-Fock (or
Thomas-Fermi) theory. Let us introduce the spacial density of the particle
with the state 𝝭 ∈ H:
∫︁
(0.3.1) 𝜌(x) = 𝜌 (x) = N |𝝭(x,x ,...,x )|𝟤dx ···dx .
𝝭 𝟤 N 𝟤 N
5
Let us write the Hamiltonian, describing the corresponding “quantum liq-
uid”:
∫︁ ∫︁
𝟣
(0.3.2) ℰ (𝜌) = 𝜏 (𝜌(x))dx − V(x)𝜌(x)dx + 𝖣(𝜌,𝜌),
B B
𝟤
with
∫︁∫︁
(0.3.3) 𝖣(𝜌,𝜌) = |x −y|−𝟣𝜌(x)𝜌(y)dxdy
where 𝜏 is the energy density of a gas of noninteracting electrons:
B
(︀ )︀
(0.3.4) 𝜏 (𝜌) = 𝗌𝗎𝗉 𝜌w −P(w)
B
w≥𝟢
is the Legendre transform of the pressure P (w) given by the formula
B
(︁𝟣 d ∑︁ d)︁
(0.3.5) P (w) = 𝜘 B w𝟤 + (w −𝟤jB)𝟤
B 𝟣 𝟤 + +
j≥𝟣
with 𝜘 = (𝟤𝜋)−𝟣q,(𝟥𝜋𝟤)−𝟣q for d = 𝟤,𝟥 respectively.
𝟣
The classical sense of the second and the third terms in the right-hand
expression of (0.3.2) is clear and the density of the kinetic energy is given by
𝜏 (𝜌) in the semiclassical approximation (see remark 0.3.1). So, the problem
B
is
(0.3.6) Minimize functional ℰ (𝜌) defined by (0.3.2) under restrictions:
B
∫︁
(0.3.7) 𝜌 ≥ 𝟢, 𝜌dx ≤ N.
𝟣,𝟤
The solution if exists is unique because functional ℰ (𝜌) is strictly convex
B
(see below). The existence and the property of this solution denoted further
by 𝜌𝖳𝖥 is known in the series of physically important cases.
B
Remark 0.3.1. If w is the negative potential then
(0.3.8) 𝗍𝗋e(x,x,𝟢) ≈ P′ (w)
B
defines the density of all non-interacting particles with negative energies at
point x and
∫︁ 𝟢 ∫︁
(0.3.9) 𝜏 d 𝗍𝗋e(x,x,𝜏)dx ≈ − P (w)dx
𝜏 B
−∞
is the total energy of these particles; here ≈ means “in the semiclassical
approximation”.
6
We consider in the case of d = 𝟥 a large (heavy) molecule with potential
(24.1.4) of [Ivr11]. It is well-known4) that
Proposition 0.3.2. (i) For V(x) given by (0.1.4) minimization problem
(0.3.6) has a unique solution 𝜌 = 𝜌𝖳𝖥; then denote ℰ𝖳𝖥 := ℰ (𝜌𝖳𝖥);
B B B B
∑︀
(ii) Equality in (0.3.7) holds if and only if N ≤ Z := Z ;
𝟤 m m
(iii) Further, 𝜌𝖳𝖥 does not depend on N as N ≥ Z;
(iv) Thus
∫︁
∑︁
(0.3.10) 𝜌𝖳𝖥dx = 𝗆𝗂𝗇(N,Z), Z := Z .
B m
𝟣≤m≤M
0.4 Main results sketched and plan of the
chapter
In the first half of this Chapter we derive asymptotics for ground state energy
and justify Thomas-Fermi theory. As construction of Section 24.2 of [Ivr11]
works with minimal modifications (see Section 5) in the magnetic case as
well we start immediately from magnetic Thomas-Fermi theory in Section 1.
We discover that there are three different cases: a weak magnetic field case
B ≪ Z𝟦 when ℰ𝖳𝖥 ≍ Z𝟧 and ℰ𝖳𝖥 = ℰ𝖳𝖥(𝟣+o(𝟣)), a strong magnetic field
𝟥 B 𝟥 B 𝟢
case B ≫ Z𝟦 when ℰ𝖳𝖥 ≍ B𝟤Z𝟫 and ℰ𝖳𝖥 = ℰ̄𝖳𝖥(𝟣 + o(𝟣)) where ℰ̄𝖳𝖥 is
𝟥 B 𝟧 𝟧 B B B
Thomas-Fermi potential derived as P (w) = 𝟣𝜘 wd (cf. (0.3.5)), and an
B 𝟤 𝟣 𝟤
intermediate case B ∼ Z𝟦.
𝟥
Then we apply semiclassical methods (like in Section 24.4) of [Ivr11]
albeit now analysis is way more complicated due to two factors: semi-
classical theory of magnetic Schr¨odinger operator is more difficult than
the corresponding theory for non-magnetic Schr¨odinger operator and also
Thomas-Fermi potential W𝖳𝖥 is not very smooth in the magnetic case, so
we need to approximate it by a smooth one (on a microscale). We discover
that both semiclassical methods and Thomas-fermi theory are relevant only
as B ≪ Z𝟥. Case of the superstrong magnetic field B ≫ Z𝟥 was considered
in E. H. Lieb, J. P. Solovej and J. Yngvarsson [LSY1] and we hope to cover
𝟣 Pending it in the Chapter 27𝟣
4) Section IV of E. H. Lieb, J. P. Solovej and J. Yngvarsson [LSY2].
7
First of all, in Section 2 we consider the case M = 𝟣; then Thomas-Fermi
potential W𝖳𝖥 is non-degenerate and in this case we derive sharp spectral
B
asymptotics.
Next, in Section 3 we consider the case M ≥ 𝟤 but only in the zone
{W𝖳𝖥+𝜈 ≥ B} where 𝜈 is a chemical potential and B is an intensity of the
B
magnetic field. A certain weaker non-degeneracy condition is satisfied due
to Thomas-Fermi equation and we derive almost sharp spectral asymptotics.
Then in Section 4 we consider the case M ≥ 𝟤 in the zone {W𝖳𝖥+𝜈 ≤ B}
containing the boundary of 𝗌𝗎𝗉𝗉𝜌𝖳𝖥; this is the most difficult case to analyze
B
and our remainder estimates are not sharp unless N ≥ Z −CZ𝟤.
𝟥
Finally, in Section 5 we derive asymptotics of the ground state energy.
Their precision (or lack of it) follows from the precision of the corresponding
semiclassical results; so our results in the case M = 𝟣 are sharp, but results
in the case M ≥ 𝟤 (especially if N ≤ Z −CZ𝟤) are not.
𝟥
𝟤 Pending In the second half𝟤 of this Chapter we consider related problems. In
Section 6 (cf. Section 24.5) of [Ivr11] we consider negatively charged systems
(N ≥ Z) and estimate both ionization energy 𝖨 and excessive negative
N
charge (N −Z) 5).
+
In Section 7 (cf. Section 24.6) of [Ivr11] we consider positively charged
systems (N ≤ Z) and estimate the remainder |𝖨 +𝜈| in the formula 𝖨 ≈ −𝜈;
N N
as M ≥ 𝟤 we also estimate excessive positive charge (Z −N) when atoms
+
can be bound into molecule5).
We will assume that
(0.4.1) c−𝟣N ≤ Z ≤ cN ∀m = 𝟣,...,M.
m
1 Magnetic Thomas-Fermi theory
1.1 Framework and existence
TheThomas-Fermitheoryiswelldevelopedinthemagneticcaseaswellalbeit
inthelesserdegreethaninthenon-magneticone. Themostimportantsource
now is Section IV of E. H. Lieb, J. P. Solovej and J. Yngvarsson [LSY2].
Again as in the previous Paper [Ivr10] to get the best lower estimate
for the ground state energy (neglecting semiclassical errors) one needs to
5) In (magnetic) Thomas-Fermi theory both answers are 𝟢.
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maximize functional 𝝫 (W +𝜈) defined by (24.3.1) of [Ivr11] albeit with
B,*
the pressure P (w) given for d = 𝟤,𝟥 by (0.3.5). Formulae (24.3.2) and
B
(24.3.3) of [Ivr11] also remain valid.
Further, to get the best upper estimate (neglecting semiclassical errors)
one needs to minimize functional 𝝫*(𝜌′,𝜈) defined by (24.3.4) of [Ivr11]
B
where(24.3.4)of[Ivr11]remainsvalidwithP replacedbyP andrespectively
B
𝜏(𝜌′) replaced by 𝜏 (𝜌′) which is Legendre transformation of P (see (0.3.4)).
B B
Since P is given by much more complicated expression (0.3.5) rather
B
than (24.3.6) of [Ivr11], and respectively
𝟣
(1.1.1) P′ (w) = d𝜘 B(︁𝟣wd𝟤−𝟣 +∑︁(w −𝟤jB)d𝟤−𝟣)︁
B 𝟤 𝟣 𝟤 + +
j≥𝟣
(cf. (24.3.6) ) of [Ivr11] there is no explicit expression for 𝜏 similar to
𝟤 B
(24.3.7) of [Ivr11].
Remark 1.1.1. (i) B(x) = |∇×A(x)|;
(ii) From now on we will assume that d = 𝟥;
(iii) P is a strictly convex function and therefore 𝜏 is also a strictly convex
B B
function6);
(iv) P (w) → P (w), P′ (w) → P′(w) and 𝜏 (𝜌) → 𝜏 (𝜌) as B → 𝟢 where
B 𝟢 B 𝟢 B 𝟢
(without subscript “𝟢”) the limit functions have been defined by (24.3.6)
and (24.3.7) of [Ivr11] respectively.
Remark 1.1.2. (i) Alternatively we minimize ℰ (𝜌) = 𝝫*(𝜌,𝟢) under as-
B B
sumptions
∫︁
(1.1.2) 𝜌 ≥ 𝟢, 𝜌dx ≤ N;
𝟣,𝟤
(ii) So far in comparison with the previous paper [Ivr10] we changed only
definition of P (w) and 𝜏 (𝜌) respectively. Note that P (w) belongs to C d
B B B 𝟤
(as d = 𝟤,𝟥) as function of w; this statement will be quantified later;
6) As d =𝟤, P is a convex piecewise linear function and therefore 𝜏 is also a convex
B B
function.
9
(iii) While not affecting existence (with equality in (1.1.2) iff N ≤ Z) and
𝟣
uniqueness of solution, it affects other properties, especially as B ≥ Z𝟦.
𝟥
Proposition 1.1.3. In our assumptions for any fixed 𝜈 ≤ 𝟢 statements
(i)–(vii) of proposition 24.3.1 of [Ivr11] hold.
Proof. The proof is the same as of proposition 24.3.1 of [Ivr11]. The proof
that threshold 𝜈 = 𝟢 matches to N = Z are theorems 4.9 and 4.10 of Section
IV of E. H. Lieb, J. P. Solovej and J. Yngvarsson [LSY2].
Note that (24.3.8)–(24.3.9) and (24.3.10) of [Ivr11] become
𝟣
(1.1.3) 𝜌 = 𝝙(W −V) = P′ (W +𝜈),
𝟦𝜋 B
(1.1.4) W = o(𝟣) as |x| → ∞
and
∫︁
(1.1.5) 𝒩(𝜈) = P′ (W +𝜈)dx
B
respectively.
Similarly, proposition 24.3.2 of [Ivr11] remains true:
Proposition 1.1.4. For arbitrary W the following estimates hold with ab-
solute constants 𝜖 > 𝟢 and C :
𝟢 𝟢
(1.1.6) 𝜖 𝖣(𝜌−𝜌𝖳𝖥,𝜌−𝜌𝖳𝖥) ≤ 𝝫 (W𝖳𝖥 +𝜈)−𝝫 (W +𝜈) ≤
𝟢 B,* B,*
C 𝖣(𝜌−𝜌′,𝜌−𝜌′)
𝟢
and
(1.1.7) 𝜖 𝖣(𝜌′ −𝜌𝖳𝖥,𝜌′ −𝜌𝖳𝖥) ≤ 𝝫*(𝜌,𝜈)−𝝫*(𝜌𝖳𝖥,𝜈) ≤
𝟢 B B
C 𝖣(𝜌−𝜌′,𝜌−𝜌′)
𝟢
with 𝜌 = 𝟣 𝝙(W −V), 𝜌′ = P′ (W +𝜈).
𝟦𝜋 B
Proof. This proof is rather obvious as well.
10