a r X i v :c o n d -m a t /0610725v 1 [c o n d -m a t .s t r -e l ] 26 O c t 2006Pair density functional theory by means of the correlated wave
function
Masahiko Higuchi Department of Physics,Faculty of Science,Shinshu University,Matsumoto 390-8621,Japan Katsuhiko Higuchi Graduate School of Advanced Sciences of Matter,Hiroshima University,Higashi-Hiroshima 739-8527,Japan (Dated:February 6,2008)Abstract We present a density functional scheme for calculating the pair density (PD)by means of the correlated wave function.This scheme is free from both of problems related to PD functional theory,i.e.,(a)the need to constrain the variational principle to N -representable PDs and (b)the development of a kinetic energy functional.By using the correlated wave function,the searching region for the ground-state PD is substantially extended as compared with our previous theory[Physica B 372(2007),in press].The variational principle results in the simultaneous equations that yield the best PD beyond the previous theory,not to mention the Hartree-Fock approximation.Note:The previous paper [Physica B 372(2007),in press]can be downloaded from “Arti-cle in press”in the website of Physica B.
I.INTRODUCTION
The pair density(PD)functional theory has recently attracted particular interest because it provides the obvious way to improve on the density functional theory(DFT)[1,2,3,4,5, 6,7,8,9].Ziesche?rst proposed the PD functional theory about a decade ago[1,2],and then many workers followed his work and have developed a variety of approaches[3,4,5,6,7,8,9].
Very recently,we have proposed an approximate scheme for calculating the PD on the basis of the extended constrained-search theory[10,11,12,13,14].By introducing the noninteracting reference system[10,11],the resultant PD corresponds to the best solution within the set of the PDs that are constructed from the single Slater determinant(SSD). This PD functional theory has two kinds of merits.The?rst one is that the reproduced PD is necessarily N-representable.This is a strong merit because the necessary and su?cient conditions for the N-representable PD have not yet been known[15,16,17,18,19,20,21, 22,23,24,25,26].The second merit is the tractable form of the kinetic energy functional. The kinetic energy functional cannot exactly be written by using the PD alone.Some approximation is required to be introduced[7].In this theory,we have successfully given an approximate form of the kinetic energy functional with the aid of the coordinate scaling of electrons[10,11].
On the other hand,we also have the remaining problem in it[10].Namely,there exists the possibility that the solution might be far from the correct value of the ground-state PD.This is because the searching region of the PDs may be smaller than the set of N-representable PDs.In order to improve the PD functional theory with keeping the above-mentioned merits,we have to extend the searching region of the PDs to the set of N-representable PDs as closely as possible.At least,we had better extend the searching region beyond the set of the SSD-representable PDs[27].
In this paper,we shall employ the strategy for reproducing the PDs not by means of the SSD,but through the correlated wave function.As the correlated wave function,we adopt the Jastrow wave function that is de?ned as the SSD multiplied by the correlation function[28,29].Owing to the correlation function,the searching region substantially be-comes larger than the set of the SSD-representable PDs.Of course,the reproduced PD is kept N-representable because the PD is calculated via the Jastrow wave function that is a kind of antisymmetric wave functions.Also the second merit is not missed in the present
scheme by utilizing the result of the scaling property of the kinetic energy functional.
Organization of this paper is as follows.In Sec.II,we provide the preliminary de?nitions of various quantities that appear in the following sections.In Sec.III,by means of the variational principle with respect to the PD,we derive simultaneous equations that yield the best PD that is superior to the previous one[10].Such equations are quite tractable,the computational method of which are also proposed in Sec.III.
II.PRELIMINARY DEFINITIONS IN THE PD FUNCTIONAL THEORY
In this section we give the preliminary de?nitions which will be used in the present scheme.The PD is de?ned as the diagonal elements of the spinless second-order reduced density matrix,i.e.,
1
γ(2)(rr′;rr′)= Ψ|
and
E0=Min
E[γ(2)]
γ(2)
=E[γ(2)0],(5) whereΨ0,E0andγ(2)0are the ground-state wave function,ground-state energy and ground-state PD,respectively,and where E[γ(2)]is the total energy functional that is given by
2
E[γ(2)]=F[γ(2)]+
2
|r?r′|+
3,(9) where K is an arbitrary constant.
III.SINGLE-PARTICLE EQUATIONS
Equation(5)corresponds to the variational principle with respect to the PD.The search-ing region of the PDs should be of course within the set of N-representable PDs.For that
purpose,we shall introduce the searching region of the PDs that are calculated from the correlated wave functions.The searching region is substantially extended as compared with the previous theory[10],because it is restricted within the set of SSD-representable PDs. Extension of the searching region can be regarded as one of appropriate developments of the PD functional theory[27].
In this paper we adopt the Jastrow wave function as the correlated wave function.The explicit evaluation of the PD using the Jastrow wave function is actually very hard[28,29]. As a consequence,the approximation technique to evaluate the PD has been developed espe-cially in the?eld of nuclear physics.The expectation value of the PD operator with respect to the Jastrow wave function can be systematically expressed with the aid of the Yvon-Mayer diagrammatic technique[28,29].Here we shall use the lowest-order approximation of the expectation value of the PD operator.
The Jastrow wave function is de?ned as[28,29]
ΨJ(x1,x2,·····,x N)=1
C N 1≤i whereΦSSD(x1,x2,·····,x N)is the SSD,and where f(r ij)=f(|r i?r j|)is the correlation function,and where C N is the normalization constant.Suppose that the correlation function is chosen to satisfy the cusp condition for the antisymmetric wave function.The lowest-order approximation for the expectation value of the PD operator is given by[28] γ(2)(rr′;rr′)=|f(|r?r′|)|2γ(2)SSD(rr′;rr′),(11) whereγ(2)SSD(rr′;rr′)is the expectation value of the PD operator with respect to the SSD. Supposing N orthonormal spin orbitals of the SSD are denoted as{ψμ(x)},then Eq.(11) is explicitly expressed as γ(2)(rr′;rr′)= 1 where Eqs.(8),(9)and(12)are used in the?rst term on the right-hand side.The minimizing conditionδ?[{ψμ}]=0immediately leads to ν d x1{ψ?ν(x1)ψν(x1)ψμ(x)?ψ?ν(x1)ψν(x)ψμ(x1)} ×|f(|r?r1|)|2 4K3+e2N?1(v ext(r)+v ext(r1)) = νεμνψν(x),(14) where the chain rule for the functional derivatives is utilized.The Lagrange multipliersεμνshould be determined by requiring that the spin orbitals are orthonormal to each other: ψ?μ(x)ψν(x)d x=δμν.(15) In the similar way to the previous theory[10],we can simplify the above equations by means of a unitary transformation of the spin orbitals.It is easily shown thatεμνforms the Hermitian matrix.Suppose that the unitary matrix which diagonalizesεμνis written by Uμν,then i,j U?iμεij U jν=?εμδμν(16) is satis?ed,where?εμis the diagonal element of the diagonal matrix.Let us consider the following transformation of the spin orbitals: ψμ(x)= νUμνχν(x).(17) Substituting Eq.(17)into Eq.(14),and using Eq.(16),we obtain ν d x1{χ?ν(x1)χν(x1)χμ(x)?χ?ν(x1)χν(x)χμ(x1)} ×|f(|r?r1|)|2 4K3+e2N?1(v ext(r)+v ext(r1)) =?εμχμ(x).(18) Also,Eq.(15)is transformed into χ?μ(x)χν(x)d x=δμν.(19) Here note that the expression forγ(2)(rr′;rr′)in Eq.(18)is kept invariant under the unitary transformation.This is con?rmed by substituting Eq.(17)into Eq.(12),i.e., 1 γ(2)(rr′;rr′)= Equations(18)and(19)are the simultaneous equations,and the solutions yield the best PD within the set of PDs that are calculated from Eq.(20). Our previous work may be the?rst proposal of a computational approach that deals with problems related to the PD functional theory[10].The present scheme is also a computational approach,and further improves on the previous theory concerning the searching region of the PDs.In that sense,it would be useful to consider a computational procedure for solving the simultaneous equations(18)and(19). The procedure proposed here is similar to that of the Hartree-Fock equation[32].In order to make the computational procedure readily comprehensible,let us rewrite Eq.(18)as {F(r)??εδ}χδ(x)=Gδ(x)(21) with F(r)= d x1|f(|r?r1|)|2N ν=1|χν(x1)|2 × 4K3+e2N?1(v ext(r)+v ext(r1)) ,(22) Gδ(x)= d x1|f(|r?r′|)|2 N ν=1χ?ν(x1)χδ(x1)χν(x) × 4K3+e2N?1(v ext(r)+v ext(r1)) ,(23) where the spin orbitalχδ(x)is the solution of Eq.(21),and should be determined in a self-consistent way.Here note that the right-hand side of Eq.(21)comes from the second term of Eq.(20),and explicitly depends on the spin orbitalχδ(x).The key point to get the self-consistent solution is that spin orbitals of the previous iteration are used in calculating F(r)and Gδ(x)[32].By solving simultaneously Eqs.(19)and(21)with this technique,we can get a new set of spin orbitals and energy parameters?εδ’s.We continue such a procedure until the self-consistency for the solutions is accomplished[32]. IV.CONCLUDING REMARKS In this paper,we propose the PD functional theory that yields the best PD within the set of PDs that are constructed from the correlated wave https://www.doczj.com/doc/0215718315.html,pared to the previous one[10,11],the present theory has the following features. 1.The present theory is superior to the previous one in that the searching region of the PDs is certainly larger than the set of SSD-representable PDs without missing the merits of the previous theory[10].This means that the resultant PD is more reasonable than that of the previous theory. 2.The predominance of the present scheme can also be shown from the viewpoint of the total energy.If the correlation function is chosen to be unit,then the present theory is reduced to the previous one exactly.It has been already proved that the total energy of the previous theory is better than that of the Hartree-Fock approximation[11]. If the correlation function is chosen most appropriately,then the searching region is substantially equivalent to the set of PDs which are calculated by varying both correlation function and spin orbitals in Eq.(20).Therefore,the total energy of the present scheme is necessarily more sound than the previous one[10],and needless to say,than that of the Hartree-Fock approximation. 3.In addition to the above merits,the present scheme has the feature that deserves special emphasis.Due to the fact that the PD functional theory is still a developing ?eld,there hardly exist the computational approaches so far.Our previous work is perhaps the?rst paper to propose a computational approach that incorporates both of problems related to PD functional theory[10,11].The present scheme is also a computational approach.The resultant simultaneous equations are quite tractable, as well as the previous one[10,11].Also from such a viewpoint,the present scheme seems to be valuable. Thus,the resultant simultaneous equations(18)and(19)yield the PD which is de?nitely closer to the ground-state PD than the previous theory[10].Next step is to perform the actual calculations so as to con?rm to what extent the present scheme covers the N-representable PDs. Finally,we would like to comment on the future prospect of the present theory.Although the present scheme utilizes the lowest-order approximation of the expectation value of the PD operator,the higher-order corrections can proceed systematically with the aid of the Yvon-Mayer diagrams[28,29].Of course,it is anticipated that the equations will become more complicated.But,from the methodological point of view,it is important that the theoretical framework has the potentiality to improve the approximation systematically. 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[32]For instance,see,S.E.Koonin,Computational Physics(Addison-Wesley,NY,1986)Chap.3. 【精编】皮下脂肪瘤介绍及治疗方法（含民间偏方） 1 皮下脂肪瘤皮下脂肪瘤在中医称为痰核。“肉瘤”之名出《干金要方》。多因郁滞伤脾，痰气凝结所致。以皮下肉中生肿块，大如桃、拳，按之稍软，无痛为主要表现的瘤病类疾病。最常见的好发部位为颈，肩，背，臀和乳房是起源于脂肪组织的良性肿瘤，由成熟的脂肪组织所构成。 1 疾病简介皮下脂肪瘤（lipoma）是脂肪组织的良性肿瘤。由成熟的脂肪组织所构成，凡体内有脂肪存在的部位均可发生。脂肪瘤有一层薄的纤维内膜，内有很多纤维索，纵横形成很多间隔，最常见于颈、肩、背、臀和乳房及肢体的皮下组织，面部、头皮、阴囊和阴唇，其次为腹膜后及胃肠壁等处；极少数可出现于原来无脂肪组织的部位。如果肿瘤中纤维组织所占比例较多，则称纤维脂肪瘤。 2 疾病分类根据脂肪瘤的可数目可分为有孤立性脂肪瘤及多发性脂肪瘤二类。此类肿瘤好发于肩、背、臀部、四肢、腰、腹部皮下及大腿内侧，头部发病也常见。位于皮下组织内的脂肪瘤大小不一，大多呈扁圆形或分叶，分界清楚；边界分不清者要提防恶性脂肪瘤的可能。单个称为孤立行型脂肪瘤。两个或两个以上的称为多发性脂肪瘤。 按部位不同可分为皮下脂肪瘤和血管平滑肌脂肪瘤（又称错钩瘤）。根据脂肪瘤发生的部位皮下脂肪瘤为扁平或分叶状、质软，边界清楚的皮下限局性肿物。质软，可推动，表面皮肤正常，发展慢，数目多达数百个，常在皮下。血管平滑肌脂肪瘤错钩瘤多发生于各个器官（肾脏，肝脏较为多见）的毛细血管的平滑肌组织之间的脂肪瘤（又称肾错构瘤，肝错钩瘤）。 3 发病原因皮下脂肪瘤指“脂肪瘤致瘤因子”在患者体细胞内也存在一种致瘤因子，在正常情况下，这种致瘤因子处于一种失活状态（无活性状态），皮下脂肪瘤正常情况下是不会发病，但在各种内外环境的诱因影响作用下，这种脂肪瘤致瘤因子的活性处于活跃状态具有一定的活性，在机体抵抗力下降时，机体内的淋巴细胞、单核吞噬细胞等免疫细胞对致瘤因子的监控能力下降，再加上体内的内环境改变，慢性炎症的刺激、全身脂肪代谢异常的诱因条件下，脂肪瘤致瘤因子活性进一步增强与机体的正常细胞中某些基因片断结合，形成基因异常突变，使正常的脂肪细胞与周围的组织细胞发生一种异常增生现象，导致脂肪组织沉积有关，并向体表或各个内脏器官突出的肿块，称之脂肪瘤。[1] 4 发病机制皮下脂肪瘤是相当常见的皮肤病灶，由正常脂肪细胞集积而成，占软组织良性肿瘤的 80%左右，无明显特殊病因，常发于皮 精编】皮下脂肪瘤介绍及治疗方法（含民间偏方） 1皮下脂肪瘤皮下脂肪瘤在中医称为痰核。“肉瘤”之名出《干金要方》。多因郁滞伤脾，痰气凝结所致。以皮下肉中生肿块，大如桃、拳，按之稍软，无痛为主要表现的瘤病类疾病。最常见的好发部位为颈，肩，背，臀和乳房是起源于脂肪组织的良性肿瘤，由成熟的脂肪组织所构成。 1 疾病简介皮下脂肪瘤（lipoma ）是脂肪组织的良性肿瘤。由成熟的脂肪组织所构成，凡体内有脂肪存在的部位均可发生。脂肪瘤有一层薄的纤维内膜，内有很多纤维索，纵横形成很多间隔，最常见于颈、肩、背、臀和乳房及肢体的皮下组织，面部、头皮、阴囊和阴唇，其次为腹膜后及胃肠壁等处；极少数可出现于原来无脂肪组织的部位。如果肿瘤中纤维组织所占比例较多，则称纤维脂肪瘤。 2疾病分类根据脂肪瘤的可数目可分为有孤立性脂肪瘤及多发性脂肪瘤二类。此类肿瘤好发于肩、背、臀部、四肢、腰、腹部皮下及大腿内侧，头部发病也常见。位于皮下组织内的脂肪瘤大小不一，大多呈扁圆形或分叶， 分界清楚；边界分不清者要提防恶性脂肪瘤的可能。单个称为孤立行型脂肪瘤。两个或两个以上的称为多发性脂肪瘤。 按部位不同可分为皮下脂肪瘤和血管平滑肌脂肪瘤（又称错钩瘤）。 根据脂肪瘤发生的部位皮下脂肪瘤为扁平或分叶状、质软，边界清楚的皮下限局性肿物。质软，可推动，表面皮肤正常，发展慢，数目多达数百个， 常在皮下。血管平滑肌脂肪瘤错钩瘤多发生于各个器官（肾脏，肝脏较为多见）的毛细血管的平滑肌组织之间的脂肪瘤 又称肾错构瘤，肝错钩瘤）。 3发病原因皮下脂肪瘤指“脂肪瘤致瘤因子”在患者体细胞内也存在一种致瘤因子，在正常情况下，这种致瘤因子处于一种失活状态 无活性状态），皮下脂肪瘤正常情况下是不会发病，但在各种内外环境的诱因影响作用下，这种脂肪瘤致瘤因子的活性处于活跃状态具有一定的活性，在机体抵抗力下降时，机体内的淋巴细胞、单核吞噬细胞等免疫细胞对致瘤因子的监控能力下降，再加上体内的内环境改变，慢性炎症的刺激、全身脂肪代谢异常的诱因条件下，脂肪瘤致瘤因子活性进一步增强与机体的正常细胞中某些基因片断结合，形成基因异常突变，使正常的脂肪细胞与周围的组织细胞发生一种异常增生现象，导致脂肪组织沉积有关，并向体表或各个内脏器官突出的肿块，称之脂肪瘤。[1] 4发病机制皮下脂肪瘤是相当常见的皮肤病灶，由正常脂肪细胞集积而 如对您有帮助，可购买打赏，谢谢 眼角长了个小疙瘩怎么办 导语：眼睛长了个小疙瘩是很常见的现象，在这样的情况之下，就好好的进行观察，自己的小疙瘩是什么样的，然后及时的去医院做检查，及时的检查，及 眼睛长了个小疙瘩是很常见的现象，在这样的情况之下，就好好的进行观察，自己的小疙瘩是什么样的，然后及时的去医院做检查，及时的检查，及时的治疗，这是非常不错的，这样就会更好的让自己的身体恢复健康，这样是非常的不错的，那就好好的进行治疗就好了。 应该是麦粒肿的症状，别担心，初起有眼睑痒、痛、胀等不适感觉，之后以疼痛为主，少数病例能自行消退，大多数患者逐渐加重。检查见患处皮肤红肿，触摸有绿豆至黄豆大小结节，并有压痛。如果病变发生在近外眼角处，肿胀和疼痛更加明显，并伴有附近球结膜水肿。部分患者在炎症高峰时伴有恶寒发热、头痛等症状。外麦粒肿3～5日后在皮肤面，内麦粒肿2～3日后在结膜面破溃流脓，炎症随即消退。也有部分麦粒肿既不消散，也不化脓破溃，硬结节长期遗留者。它是一种没有明显疼痛的表面皮肤隆起，主要是眼睑腺体阻塞造成的伴有炎症的话才会自觉的疼痛的症状。小的霰粒肿可以热敷或者理疗按摩疗法，促进消散吸收，较大大的或长时间不能吸收的可以择期手术治疗. 这是脂肪粒，等长时间长了，熟了自己挤就行。先说说脂肪粒的产生（从其他地方转来D，自己还没那么厉害，呵呵 1、体内原因：眼部、面部出现油脂粒大多是由于近期身体内分泌有些失调，致使面部油脂分泌过剩，再加上皮肤没有得到彻底清洁干净，导致毛孔阻塞，很快形成脂肪粒。2、外在因素：眼睛周围是全身最薄的皮肤，约0.07毫米，又没有皮下腺，加上人眼一天频繁眨动，特别容易缺水、干燥、 预防疾病常识分享，对您有帮助可购买打赏 皮下组织有疙瘩是怎么回事 在一般的情况下，大家对身体健康和皮肤护理都特别的主要，尤其是在夏天的时候，皮肤都暴露在外面，接收来自外界的各种刺激，在这样的时候更要做好对皮肤的护理。有时候我，我们会发现在皮肤的下面长出来好多的小疙瘩，这样时候就会让我们感觉到不知道是怎么回事也不知道要怎么治疗。 ★考虑是毛周角化症 毛周角化症俗称“鸡皮肤”，是一种先天遗传病，个人症状 有轻有重。身体缺乏维生素A皮肤干燥，或者天气较为干燥寒冷，毛口收缩的时候腿上汗毛长不出来，症状较明显，夏季则症状较轻。一般不痛不痒，过度清洁时也可能出现瘙痒情况。 ★对策： 毛孔角化症属于先天遗传病，并不能根治，只能通过利用A 酸，或含果酸、乳酸等成分的产品保养皮肤，改善皮肤的角化程度。可去药店咨询医生购买水杨酸类软膏。 ★缺乏维生素A加重小红疙瘩 皮肤下面有小红疙瘩除了与天生体质有关以外，缺乏维生素A引起全身性的皮肤干燥也是加重症状的原因之一。 ★对策 1、洗澡不要用太热的水。过热的水会让皮肤上的皮脂过分流失，这样会加重皮肤干燥。 2、洗澡后抹身体乳。保证皮肤水分充足能有效减轻腿上小红疙瘩，所以可以在每次洗澡后涂上身体乳液。 ★可能是闭头粉刺 如果是额头、下巴等部位皮肤下有疙瘩，刚开始是黄白色，不痛不痒，天气炎热脸上油脂分泌旺盛时会有发红，多是闭头粉刺。 ★对策 1、及时清洁皮肤，注意补水，不要使用刺激性护肤品，不要用过于油腻滋润的面霜。 2、多运动促进皮肤排汗，使毛孔中的油脂脏污随着汗水排出，防止油脂堵塞毛孔发炎。 3、多喝水、多吃蔬果，饮食清淡。 4、必要时看医生。 ★可能是脂肪粒 如果皮下的疙瘩是乳白色的，不痛不痒，不受天气、温度等影响，用手不容易挤出，且多分布在眼角周围，考虑是脂肪粒。 ★对策 1、脂肪粒多是皮肤营养过剩引起的，建议不要使用太油腻的护肤品。 2、可取美容院用针挑破，挤出脂肪粒。(别用手挤，小心皮肤感染发炎)皮下脂肪瘤介绍及治疗方法
完整版皮下脂肪瘤介绍及治疗方法含民间偏方
眼角长了个小疙瘩怎么办
皮下组织有疙瘩是怎么回事
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