To the theory of interaction between electron and nuclear systems

英语教学综合专硕考研王蔷《英语教学法教程》考研真题集一、语言和语言学习Ⅰ. Fill in the blanks.1. According to the _____ theory represented by Vygotsky, learning is best achieved through the dynamic interaction between the teacher and the learner and between learners.（山东师范大学2018研）【答案】socio-constructivist @~！【解析】本题考查的是社会建构主义理论的观点。

以维果茨基为代表的社会建构主义理论认为学习的最佳途径是师生之间以及学生之间的活跃互动。

2. The functional view not only sees language as a _____ system but also a means for doing things.（安徽师范大学2015研）【答案】linguistic @~！【解析】本题考查的是功能主义语言理论。

该种语言理论不仅将语言视作语言系统，还将它看作一种方法。

3. There are at least three theoretical views of language and the nature of language proficiency. The first is ______, the second is ______ and the third is ______.（天津师范大学2012研）【答案】structural view，functional view，interactional view @~！【解析】本题考查的是语言理论。

语言理论有三种，分别是结构主义语言理论、功能主义结构理论和交互式语言理论。

4. The constructivist theory believes that _____ is a process in which learners construct meaning based on their own experiences and what they already know.（山东师范大学2017研）【答案】learning @~！【解析】本题考查的是建构主义学习理论的观点。

行为主义学习理论Behavioral learning theory20世纪20年代，华生〔J·B Watson〕创立了行为主义学说。

这种理论认为，人类的一切行为，其构成的根本要素是反响，一切行为表现只是多种反响的组合；而该等反响中，除少数是生而具有的反射之外，全都是个体在适应环境时，与其环境中各个刺激之间的关系，经由经典条件作用的学习过程所形成的。

因此，只要能理解环境刺激与个体的关系，就可以设计并控制刺激，经由条件作用的方法，建立起所要建立的反响，从而组成预期的复杂行为；而且，也可根据条件作用法那么〔指消弱〕，消除个体已有的行为。

1920s, Watson (J.B Watson) founded Behaviorism. This theory holds that all human behavior, which constitute the basic elements of the reaction, is a combination of a variety of reactions; and such reactions, except for a few are born with reflection outside are all individual in adapting environment, their environment between the various stimuli, through the learning process of classical conditioning formed. Therefore, as long as understand the relationship between environmental stimuli and the individual, we can design and control the stimulus through conditioning methods, to establish the response you want to create, and thus the expected composition of complex behavior; but also the role of law in accordance with the conditions (refer to eliminate weak), eliminating the existing behavior of the individual.行为主义理论又称刺激——反响(S—R)理论，是当今学习理论的主要流派之一。

理论和实际的联系英语作文1. Theory and practice are like two sides of the same coin. Without theory, practice lacks direction and purpose; without practice, theory remains abstract and disconnected from reality.2. The connection between theory and practice can be seen in everyday life. For example, learning the theory of cooking may help you understand the science behinddifferent cooking techniques, but it is only through practice that you can truly master the art of cooking delicious meals.3. In the field of education, teachers often face the challenge of bridging the gap between theory and practice. They must not only understand educational theories but also apply them effectively in the classroom to meet the diverse needs of their students.4. The connection between theory and practice is alsoevident in the world of business. A business theory may suggest a certain marketing strategy, but it is only through practical implementation and adaptation that businesses can achieve success in the competitive market.5. In the field of medicine, doctors rely on both theoretical knowledge and practical experience to diagnose and treat patients. Theoretical understanding of medical conditions is essential, but it is the hands-on experience of treating patients that helps doctors develop theirskills and expertise.6. Ultimately, the connection between theory and practice is essential for progress and innovation in all fields. It is through the integration of theory and practice that new ideas are born, existing knowledge is expanded, and real-world problems are solved.。

The Interaction Between Light andMatter光和物质的相互作用光和物质的相互作用是自然界中的一个重要现象，有着广泛而深刻的影响。

光是电磁波，而物质则是由原子、离子或分子组成的。

当光与物质相互作用时，就会发生很多有趣的现象，如折射、反射、散射、吸收和发射等。

这些现象既有理论上的解释，也可以应用到很多领域中。

折射和反射折射和反射是光线与界面相互作用时的两个主要现象。

当光线从一种介质射向另一种介质时，会改变传播方向，这种现象称为折射。

折射的角度与入射角度之比称为折射率，其大小取决于两种介质的物理性质。

反射是指光线遇到平滑的表面后，从表面反弹回来的现象。

反射的角度等于入射角度，这是由光的传播规律所决定的。

反射还有很多应用，如光学反射镜、光学望远镜等。

散射和吸收散射是指光线穿过介质时发生方向改变的现象，通常由于介质中存在的微观杂质、气体或液滴等而产生。

散射还可以发生在分子或原子的表面上，这种现象被称为拉曼散射。

散射还有很多应用，如激光散射测量、光学成像等。

吸收是指介质对光线的一部分或全部能量吸收的现象。

当光线被吸收后，其能量可以被转化成热能或化学能，从而对物质的性质和状态产生影响。

吸收还有很多应用，如光能电池、光催化等。

发射和激发发射是指当物质受到能量激发后，会放出能量的一部分或全部，这种现象称为发射。

发射可以是自发的，也可以是受到外部刺激后发生的。

自发发射通常包括热辐射和荧光等现象，而受外部刺激后的发射则包括激光和电致发光等现象。

发射现象在光谱学、光化学和光学传感等领域中有广泛的应用。

激发是指物质受到外部刺激后，能级发生变化，从低能级向高能级跃迁，发生吸收或发射现象。

激发还有很多应用，如荧光定量分析、激光切割等。

总结光和物质的相互作用是自然界中的一种基本现象，涉及到光学、化学、材料科学等多个领域。

了解光和物质的相互作用规律，可以为科学研究和技术开发提供重要的理论支持和实验方法。

Theory and Practice of Science and Technology2022, VOL. 3, NO. 6, 118-122DOI: 10.47297/taposatWSP2633-456919.20220306Research on Interactive Model of Curriculum Standards Implementation -- Taking Java Programming as an Example Guohua XiongGuangdong Construction Polytechnic, Guangzhou Guangdong 510440, ChinaABSTRACTIn order to improve the teaching quality of software technology specialty,this paper first analyzes the foreign interactive teaching model,reconstructs the theoretical analysis framework for teachers to implementcurriculum standards, summarizes the core indicators for teachers toimplement curriculum standards and designs a suitable curriculumstandard implementation model, and verifies the rationality of the modelwith the implementation of the curriculum standard of JavaProgramming as an example, Through the joint implementation ofteachers and students, the results show that the model can effectivelyimprove students' learning interest and practical ability, and can provide atheoretical basis for promoting teachers' professional development in thenew curriculum and teaching reform.KEYWORDSCurriculum standard; Interactive model; Teaching implementation; JavaProgramming1　IntroductionThe implementation of curriculum standards carries educational ideas, is a professional and practical work, and has complex value orientation. How to effectively implement curriculum standards and promote teacher professionalization is the key to the realization of curriculum teaching design. The implementation of curriculum standards has always been put in a prominent position in the curriculum reform research of all countries in the world. How to inspire teachers to think about the implementation process of curriculum standards "like experts" as a whole [1], pay attention to students' learning and growth, and form a good teacher-student interaction has become a hot research issue for educators.2　Research Status(1)　Research status abroadForeign scholars pay attention to the study of the interaction between teachers and curriculum, and the representative researchers are Remilard, Drake, etc. They are mainly based on teachers' empirical research to understand the interaction between teachers and curriculum, and use this as an analytical framework to solve the problem of teachers' implementation of curriculum standards. They regard the development of theoretical models as an important part of the implementation of curriculum standards, and the number of models constructed is complex. From the perspective of research, the model can be divided into two categories. One is the model of teachers' curriculumTheory and Practice of Science and Technology implementation, such as the famous "Concern-based adoption model (CBAM)"; The other is the interactive model for the implementation of curriculum standards, such as the curriculum hierarchy theory created by Goodlad in the early stage, and the "teacher student content" interactive model developed by Anderson on the basis of this theory.(2)　Domestic research statusThe research in China is relatively late. The research on teachers' implementation of curriculum standards mainly focuses on the implementation level, influencing factors and effectiveness. It is difficult to explore the development context of the implementation of curriculum standards, and the theoretical depth is not enough. Although the theoretical research on the implementation of curriculum standards by teachers has been continuing in recent years, it is still in the exploration period, and there is no systematic theory of curriculum implementation. Some domestic scholars equate the implementation of curriculum standards with teaching, and "curriculum" is regarded as teaching materials. The implementation of curriculum standards is to implement the content of teaching materials; Another part of scholars believe that the implementation of curriculum standards is the process of putting innovative curriculum plans into practice [2].3　Research IdeasBy comparing the research on the implementation of curriculum standards by teachers at home and abroad, it is found that there are some typical models for the implementation of curriculum standards by foreign teachers, and they have developed in depth; However, in China, there are either speculative discussions or direct practical explorations, and there are few theoretical systems for teachers to implement curriculum standards. In view of this, the research idea of this paper is to reconstruct the theoretical analysis framework of teachers' implementation of curriculum standards through the theoretical analysis of foreign typical models, and on this basis, summarize the key issues of teachers' implementation of curriculum standards and design a suitable curriculum standard implementation model, and take the implementation of Java Programming curriculum standards as an example to verify the rationality of the model, At the same time, the consistency of teaching evaluation in the implementation of curriculum standards by teachers is analyzed, and thespecific ideas are shown in Figure 1.Figure 1　Research ideas 119Guohua Xiong 1204　Construction of Interactive Model for Curriculum Standard Implementation (1)　Concept of interactive teaching modeThe literature [3]believes that the interactive teaching mode refers to the teaching principle of "students as the main body and teachers as the leading role", which focuses on heuristic teaching and allows students and teachers to participate in the teaching and learning of the course. Literature [4] believes that it is a teaching structure model aimed at cultivating students' independent consciousness and innovation ability, and at "making students love, learn and be good at learning". This paper believes that the interactive teaching mode refers to the teaching mode that, under the guidance of the teaching ideology of "students as the main body, teachers as the leading, and training projects as the main line", around the training projects, interactive teaching methods, supplemented by heuristic methods, let teachers and students participate in the bilateral activities of training project teaching and learning, so as to achieve the goal of cultivating students' innovative spirit and improving practical ability.(2)　Relevant theoretical basisAt present, the relevant theories of interactive teaching research mainly focus on constructivism and humanism.Constructivism is one of the most influential educational concepts in the world today. It puts forward a series of new explanations for learning and teaching. It believes that learning is carried out under specific social and cultural background and specific learning environment conditions. When designing practical projects, combining the actual situation of real life is conducive to stimulating students' interest and overcoming the abstraction of learning content To enable students to fully use their knowledge and experience in their cognitive structure, give play to their association and imagination, and promote the effective transfer of students' knowledge [5] [6]. The essence of constructivism is to analyze the nature of cognitive activities and put forward a new understanding of the construction process of learning. Constructivism regards students as the subject and center of the teaching process, attaches importance to the role of students' existing knowledge and experience and psychological structure, attaches importance to the subjective initiative of students' learning process, emphasizes the social and situational nature of students' learning, and believes that teachers should play the role of organizers, helpers and promoters in the teaching process, reflecting the student-oriented teaching concept.Humanism education theory mainly emphasizes the teaching goal view of knowing and unifying, the meaningful free learning view and the student-centered teaching view. Its emphasis is not on the teaching results but on the teaching process, emphasizing the development of students' potential, personality and creativity in the process; What we pay attention to is not the teaching content but the teaching method. In the teaching method, we advocate student-centered, everything for the development of students, and let students choose and discover themselves.(3)　Curriculum standard interactive model designSince the teaching of the curriculum follows the corresponding curriculum standards, a better interactive curriculum standard is, to a certain extent, the prerequisite for achieving high-quality curriculum interaction. For another course, from the beginning to the end of the course assessment, the interaction between teachers and students is permeated in the whole teaching process, so we divide the design of interactive curriculum standards into three stages.The first stage is to interact before class and formulate curriculum standards. Before each class,Theory and Practice of Science and Technology the teacher requires the students to preview. Before class, the teacher should explain the teaching objectives, key and difficult points of the course, relevant requirements of practical training and scoring standards to the students. Then, through questionnaires, group discussions, network tool interaction and other methods, master the students' preview situation and basic needs for the course content, and carry out pre-class learning situation analysis based on this, and then modify or redesign in combination with the original curriculum standard to develop a new curriculum standard. Take part of the contents of Chapter 2 Java Programming Basics of Java Programming for example, as shown in Table 1. Since students have learned C Content Language Programming before learning this course, they use the form of pre-class questionnaire to achieve the purpose of interaction. This can clarify what content students can learn by themselves, what content needs to be taught and what kind of teaching method students can accept more easily. Based on this, the curriculum standard of this course can be preliminarily designed.The second stage is interaction in class and adjustment of curriculum standards. During the course of teaching according to the preliminarily designed curriculum standard, analyze whether the students' mastery of knowledge is consistent with the pre-class analysis according to the students' actual training completion and the students' performance in the discussion link. During the course of teaching, students may also put forward some new questions. At this time, the teacher should make a good record and analysis of the learning situation in class, and modify the preliminarily implemented curriculum standard after class according to the analysis. Java Program Design is a practical course, which uses students to achieve the organic integration of theoretical knowledge and practical skills. The course selection is based on project-based teaching. In class, teachers and students jointly discuss the needs analysis of the project. The key technology realization is explained by teachers in the way of theoretical teaching and practical demonstration, and then by students on the computer, To reflect the student-centered concept, students can also design and complete independently.The third stage is after-class interaction and optimization of curriculum standards. After class, teachers should not only organize students to investigate the teaching satisfaction of this class, but also ask students to put forward good practices and problems that need to be improved, and analyze students' mastery of the knowledge of this class according to the completion of homework after class, so as to further optimize the curriculum standards of this class.Through the above theoretical analysis, the designed interactive model for the implementation of curriculum standards is shown in Figure 2.Table 1　Analysis of basic knowledge points of Java programmingKnowledge PointsBasic format of Java programComments in JavaIdentifier in JavaKeywords in JavaConstants in JavaDefinition of variablesData type of variableType conversion of variableScope of variableArithmetic operatorAssignment OperatorsComparison operatorLogical operatorOperator precedence Self-study or not √√√√√√√Key Lectures √√√√√√√Teaching Method Lecture and Demonstration Lecture and Demonstration Lecture and Demonstration Lecture and Demonstration Lecture and Demonstration Lecture and DemonstrationLecture and Demonstration 121Guohua Xiong 5　ConclusionAt present, there is still a lack of research on the interactive model of curriculum standard implementation, and even less empirical research and specific teaching practice are involved. Starting from this weak link, this paper comprehensively introduces the concept, basis and process of the establishment of interactive curriculum standards in three stages of pre-class, in-class and post-class, and accordingly designs an interactive model for the implementation of curriculum standards, and validates the model through the implementation of the curriculum standard Java Programming. The results of the validation not only deepen the teachers' understanding of the implementation process of curriculum standards to a certain extent, More importantly, it can provide some theoretical basis for promoting teachers' professional development in the new curriculum and teaching reform, and provide a framework for reference and analysis for improving teachers' professional practice.FundingThis paper is Supported by Educational Planning Project of Guangdong Province in 2021 (Grant No. 2021GXJK534), Supported by Project of China Construction Education Association in 2021 (Grant No.2021177), Supported by Science and Technology Innovation Strategy Special Project in Guangdong Province in 2022 (Grant No.pdjh2022b0834), Supported by universities characteristic innovation project of Guangdong Province in 2022 (Grant No. 2022KTSCX248), Supported by School level project of Guangdong Construction Polytechnic in 2021 (Grant No.JG2021-12).References[1] Research on the interactive model of teachers'implementation of curriculum standards[D].Yangtze University,2020.[2] XIA Xuemei,CUI Yunhuo.Interaction Perspective of Teachers'Curriculum Implementation:A Construction Based on Literature Review of 40 Years[J]. Research in Educational Development,2013,33(24):1-5+10.[3] Zhao Liqun. The Research and Practice of Interactive Pattern in Class between the Teacher and Students[D]. Liaoning Normal University,2006.[4] Li Ment. Instructional Pattern Construction and Practice Research of the Synchronous Interactive Class[D]. Central China Normal University,2015.[5] ZHANG Zhiliang, WANG Meng, LI Hui. Research on experimental teaching based on constructivism learning theory[J]. Laboratory Science. 2022,25(04):93-96.[6] Zhang Sai.Integrating Innovation Education into Professional Education Practice Based on Anchoring Teaching Model:Taking Changjiang Polytechnic as an Example[J]. Journal of Hubei Adult Education Institute. 2022,28(03):62-66.Figure 2　Interactive model of curriculum standard implementation 122。

行为主义学习理论Behavioral learning theory20世纪20年代，华生（J·B Watson）创立了行为主义学说。

这种理论认为，人类的一切行为，其构成的基本要素是反应，一切行为表现只是多种反应的组合；而该等反应中，除少数是生而具有的反射之外，全都是个体在适应环境时，与其环境中各个刺激之间的关系，经由经典条件作用的学习过程所形成的。

因此，只要能了解环境刺激与个体的关系，就可以设计并控制刺激，经由条件作用的方法，建立起所要建立的反应，从而组成预期的复杂行为；而且，也可根据条件作用法则（指消弱），消除个体已有的行为。

1920s, Watson (J.B Watson) founded Behaviorism. This theory holds that all human behavior, which constitute the basic elements of the reaction, is a combination of a variety of reactions; and such reactions, except for a few are born with reflection outside are all individual in adapting environment, their environment between the various stimuli, through the learning process of classical conditioning formed. Therefore, as long as understand the relationship between environmental stimuli and the individual, we can design and control the stimulus through conditioning methods, to establish the response you want to create, and thus the expected composition of complex behavior; but also the role of law in accordance with the conditions (refer to eliminate weak), eliminating the existing behavior of the individual.行为主义理论又称刺激——反应(S—R)理论，是当今学习理论的主要流派之一。

characterized interactions Characterized interactions refer to the various forms of exchanges and relationships between individuals, groups, or even countries that are marked by distinct traits or qualities. These interactions can take place in various spheres of life, such as personal, social, economic, or political. In this article, we will explore the concept of characterized interactions and provide a step-by-step analysis of their different aspects.Step 1: Understanding Characterized Interactions Characterized interactions are essentially interactions that are distinguished by specific traits or features. These traits can include the nature of the relationship, the intensity of the interaction, the purpose or goal of the interaction, and the impact it has on the parties involved.Step 2: Types of Characterized InteractionsThere are numerous types of characterized interactions, depending on the context in which they occur. Some common types include: 1. Personal interactions: These interactions occur between individuals on a personal level, such as family members, friends, or romantic partners. They are characterized by emotional closeness,trust, and shared experiences.2. Social interactions: These interactions occur within a society or community and can be both formal and informal. Examples include interactions at social gatherings, in educational institutions, or at workplaces. They are characterized by social norms, expectations, and hierarchies.3. Economic interactions: These interactions occur in the realm of business and trade. They involve transactions, negotiations, and collaborations between individuals, companies, or even countries. They are characterized by economic motives, competition, and mutual benefits.4. Political interactions: These interactions take place in the political arena and involve interactions between governments, political parties, or international organizations. They are characterized by power dynamics, diplomacy, and negotiation.Step 3: Factors Influencing the Characterization of Interactions Several factors can influence the characterization of interactions. These factors can shape the nature, intensity, and outcome of the interactions. Some key factors include:1. Culture: Cultural norms and values play a significant role in shaping interactions. Different cultures may prioritize differentvalues, such as individualism, collectivism, or hierarchy, which can impact the way interactions are conducted.2. Power dynamics: Power imbalances can significantly affect the characterization of interactions. Interactions between individuals or groups with unequal power may involve domination, oppression, or exploitation.3. Context: The context in which interactions take place, such as the environment, time, or specific circumstances, can have a significant impact on their characterization. For example, interactions during a crisis may be characterized by urgency or cooperation.4. Communication: Effective communication is crucial for successful interactions. The way information is exchanged, interpreted, and understood can shape the dynamics and outcomes of interactions.Step 4: Importance of Characterized Interactions Characterized interactions play a vital role in human society and have several important implications:1. Building relationships: Interpersonal interactions are essential for building and maintaining relationships. These interactions lay the foundation for trust, empathy, and mutual understanding.2. Social cohesion: Interactions within communities foster a sense of belonging and social cohesion. They allow individuals to bond,collaborate, and work towards common goals.3. Economic development: Economic interactions drive trade, innovation, and growth. They facilitate the exchange of goods, services, and knowledge, creating opportunities for economic development.4. Conflict resolution: Political interactions can be instrumental in resolving conflicts and fostering peace. Dialogue, negotiation, and diplomacy are essential tools in addressing differences and promoting cooperation.Step 5: Challenges in Characterized InteractionsWhile characterized interactions have significant benefits, they also face challenges that can hinder their effectiveness:1. Cultural differences: In a globalized world, interactions often involve individuals from diverse cultural backgrounds. These cultural differences can lead to misunderstandings or conflicts if not managed effectively.2. Power inequalities: Unequal power dynamics can create barriers to effective interactions. Marginalized groups may struggle to assert themselves or have their voices heard.3. Miscommunication: Inadequate communication can lead to misunderstandings, mistrust, or misinterpretation of intentions.Clear and effective communication is essential for successful interactions.4. Conflict of interests: Different parties may have conflicting interests or goals, leading to disagreements or competition. Effective negotiation and compromise are necessary to navigate these challenges.In conclusion, characterized interactions are a fundamental part of human life in various spheres, including personal, social, economic, and political. Understanding and effectively navigating these interactions require considering factors such as culture, power dynamics, context, and communication. While characterized interactions have several benefits, they also face challenges that need to be addressed for successful outcomes. By recognizing the importance of characterized interactions and actively working towards positive and inclusive interactions, individuals and societies can foster understanding, collaboration, and growth.。

Time-dependent Density Functional Theory Miguel A.L.Marques and E.K.U.Gross1IntroductionTime-dependent density-functional theory(TDDFT)extends the basic ideas of ground-state density-functional theory(DFT)to the treatment of excita-tions or more general time-dependent phenomena.TDDFT can be viewed an alternative formulation of time-dependent quantum mechanics but,in contrast to the normal approach that relies on wave-functions and on the many-body Schr¨o dinger equation,its basic variable is the one-body electron density,n(r,t).The advantages are clear:The many-body wave-function,a function in a3N-dimensional space(where N is the number of electrons in the system),is a very complex mathematical object,while the density is a simple function that depends solely on3variables,x,y and z.The standard way to obtain n(r,t)is with the help of afictitious system of non-interacting electrons,the Kohn-Sham system.Thefinal equations are simple to tackle nu-merically,and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential,the time-dependent Kohn-Sham potential.The exact form of this potential is unknown,and has therefore to be approximated.The scheme is perfectly general,and can be applied to essentially any time-dependent situation.Two regimes can however be observed:If the time-dependent potential is weak,it is sufficient to resort to linear-response theory to study the system.In this way it is possible to calculate,e.g.optical absorp-tion spectra.It turns out that,even with the simplest approximation to the Kohn-Sham potential,spectra calculated within this framework are in very good agreement with experimental results.On the other hand,if the time-dependent potential is strong,a full solution of the Kohn-Sham equations is required.A canonical example of this regime is the treatment of atoms or molecules in strong laserfields.In this case,TDDFT is able to describe non-linear phenomena like high-harmonic generation,or multi-photon ionization.Our purpose,while writing this chapter,is to provide a pedagogical in-troduction to TDDFT1.With that in mind,we present,in section2,a quite detailed proof of the Runge-Gross theorem[5],i.e.the time-dependent generalization of the Hohenberg-Kohn theorem[6],and the corresponding2Miguel A.L.Marques and E.K.U.GrossKohn-Sham construction[7].These constitute the mathematical foundations of TDDFT.Several approximate exchange-correlation(xc)functionals are then reviewed.In section3we are concerned with linear-response theory, and with its main ingredient,the xc kernel.The calculation of excitation en-ergies is treated in the following section.After giving a brief overlook of the competing density-functional methods to calculate excitations,we present some results obtained from the full solution of the Kohn-Sham scheme,and from linear-response theory.Section5is devoted to the problem of atoms and molecules in strong laserfields.Both high-harmonic generation and ion-ization are discussed.Finally,the last section is reserved to some concluding remarks.For simplicity,we will write all formulae for spin-saturated systems.Ob-viously,spin can be easily included in all expressions when necessary.Hartree atomic units(e=¯h=m=1)will be used throughout this chapter.2Time-dependent DFT2.1PreliminariesA system of N electrons with coordinates r∂tΨ(r,t)Ψ(r,t)|2, is interpreted as the probability offinding the electrons in positions r)+ˆW(r,t).(2) Thefirst term is the kinetic energy of the electronsˆT(r2Ni=1∇2i,(3)whileˆW accounts for the Coulomb repulsion between the electronsˆW(r2N i,j=1i=j1,t).The Hamiltonian(2)is completely general and describesTime-dependent Density Functional Theory3 a wealth of physical and chemical situations,including atoms,molecules,and solids,in arbitrary time-dependent electric or magneticfields,scattering ex-periments,etc.In most of the situations dealt with in this article we will be concerned with the interaction between a laser and matter.In that case,we can write the time-dependent potential as the sum of the nuclear potential and a laserfield,ˆV TD=ˆU en+ˆV laser.The termˆU en accounts for the Coulomb attraction between the electrons and the nuclei,ˆU en (r|r i−Rν(t)|,(5)where Zνand Rνdenote the charge and position of the nucleusν,and N n stands for the total number of nuclei in the system.Note that by allowing the Rνto depend on time we can treat situations where the nuclei move along a classical path.This may be useful when studying,e.g.,scattering experiments, chemical reactions,etc.The laserfield,ˆV laser,reads,in the length gauge,ˆVlaser(r4Miguel A.L.Marques and E.K.U.Grossthe calculation of a“simple”two electron system(the Helium atom)in a laser field takes several months in a modern computer[8](see also the work on theH+2[9]molecule and the H++3molecule[10]).The effort to solve Eq.(1)growsexponentially with the number of particles,therefore rapid developments re-garding the exact solution of the Schr¨o dinger equation are not expected.In these circumstances,the natural approach of the theorist is to trans-form and approximate the basic equations to a manageable level that still retains the qualitative and(hopefully)quantitative information about the system.Several techniques have been developed throughout the years in the quantum chemistry and physics world.One such technique is TDDFT.Its goal,like always in density-functional theories,is to replace the solution of the complicated many-body Schr¨o dinger equation by the solution of the much simpler one-body Kohn-Sham equations,thereby relieving the computational burden.Thefirst step of any DFT is the proof of a Hohenberg-Kohn type theo-rem[6].In its traditional form,this theorem demonstrates that there exists a one-to-one correspondence between the external potential and the(one-body)density.With the external potential it is always possible(in princi-ple)to solve the many-body Schr¨o dinger equation to obtain the many-body wave-function.From the wave-function we can trivially obtain the density. The second implication,i.e.that the knowledge of the density is sufficient to obtain the external potential,is much harder to prove.In their seminal paper,Hohenberg and Kohn used the variational principle to obtain a proof by reductio ad absurdum.Unfortunately,their method cannot be easily gen-eralized to arbitrary DFTs.The Hohenberg-Kohn theorem is a very strong statement:From the density,a simple property of the quantum mechani-cal system,it is possible to obtain the external potential and therefore the many-body wave-function.The wave-function,by its turn,determines every observable of the system.This implies that every observable can be written as a functional of the density.Unfortunately,it is very hard to obtain the density of an interacting sys-tem.To circumvent this problem,Kohn and Sham introduced an auxiliary system of non-interacting particles[7].The dynamics of these particles are governed by a potential chosen such that the density of the Kohn-Sham system equals the density of the interacting system.This potential is local (multiplicative)in real space,but it has a highly non-local functional de-pendence on the density.In non-mathematical terms this means that the potential at the point r can depend on the density of all other points(e.g. through gradients,or through integral operators like the Hartree potential). As we are now dealing with non-interacting particles,the Kohn-Sham equa-tions are quite simple to solve numerically.However,the complexities of the many-body system are still present in the so-called exchange-correlation(xc) functional that needs to be approximated in any application of the theory.Time-dependent Density Functional Theory5 2.2The Runge-Gross theoremIn this section,we will present a detailed proof of the Runge-Gross theorem[5], the time-dependent extension of the ordinary Hohenberg-Kohn theorem[6]. There are several“technical”differences between a time-dependent and a static quantum-mechanical problem that one should keep in mind while try-ing to prove the Runge-Gross theorem.In static quantum mechanics,the ground-state of the system can be determined through the minimization of the total energy functionalE[Φ]= Φ|ˆH|Φ .(7) In time-dependent systems,there is no variational principle on the basis of the total energy for it is not a conserved quantity.There exists,however,a quantity analogous to the energy,the quantum mechanical actionA[Φ]= t1t0dt Φ(t)|i∂2If the two potentials differ solely by a time-dependent function,they will produce wave-functions which are equal up to a purely time-dependent phase.This phase6Miguel A.L.Marques and E.K.U.Grossdensity,n(r,t),i.e.v(r,t)=v (r,t)+c(t)⇒ρ(r,t)=ρ (r,t).(9) This statement immediately implies the one-to-one correspondence between the potential and the density.In the following we will utilize primes to dis-tinguish the quantities of the systems with external potentials v and v .Due to technical reasons that will become evident during the course of the proof, we will have to restrict ourselves to external potentials that are Taylor ex-pandable with respect to the time coordinate around the initial time t0v(r,t)=∞k=0c k(r)(t−t0)k,(10)with the expansion coefficientsc k(r)=1dt kv(r,t) t=t0.(11)We furthermore define the functionu k(r)=∂k2i ∇ˆψ†(r) ˆψ(r)−ˆψ†(r) ∇ˆψ(r) .(15) We now use the quantum-mechanical equation of motion,which is valid for any operator,ˆO(t),i d∂tˆO(t)+ ˆO(t),ˆH(t) |Ψ(t) ,(16)Time-dependent Density Functional Theory7 to write the equation of motion for the current density in the primed and unprimed systemsdij (r,t)= Ψ (t)| ˆj(r),ˆH (t) |Ψ (t) .(18)dtAs we start from afixed initial many-body state,at t0the wave-functions, the densities,and the current densities have to be equal in the primed and unprimed systems|Ψ(t0) =|Ψ (t0) ≡|Ψ0 (19)n(r,t0)=n (r,t0)≡n0(r)(20)j(r,t0)=j (r,t0)≡j0(r).(21) If we now take the difference between the equations of motion(17)and(18) we obtain,when t=t0,didt k+1 j(r,t)−j (r,t) t=t0=n0(r)∇u k(r).(23) The right-hand side of Eq.(23)differs from zero,which again implies that j(r,t)=j (r,t)for t>t0.In a second step we prove that j=j implies n=n .To achieve that purpose we will make use of the continuity equation∂[n(r,t)−n (r,t)]=−∇· j(r,t)−j (r,t) .(25)∂t8Miguel A.L.Marques and E.K.U.GrossAs before,we would like an expression involving the k th time derivative of the external potential.We therefore take the(k+1)st time-derivative of the previous equation to obtain(at t=t0)∂k+2∂t k+1 j(r,t)−j (r,t) t=t0=−∇·[n0(r)∇u k(r)].(26) In the last step we made use of Eq.(23).By the hypothesis(13)we have u k(r)=const.hence it is clear that if∇·[n0(r)∇u k(r)]=0,(27) then n=n ,from which follows the Runge-Gross theorem.To show that Eq.(27)is indeed fulfilled,we will use the versatile technique of demonstra-tion by reductio ad absurdum.Let us assume that∇·[n0(r)∇u k(r)]=0with u k(r)=const.,and look at the integrald3r n0(r)[∇u k(r)]2=− d3r u k(r)∇·[n0(r)∇u k(r)](28)+ S n0(r)u k(r)∇u k(r)·d S.This equality was obtained with the help of Green’s theorem.Thefirst term on the right-hand side is zero by assumption,while the second term vanishes if the density and the function u k(r)decay in a“reasonable”manner when r→∞.This situation is always true forfinite systems.We further notice that the integrand n0(r)[∇u k(r)]2is always positive.These diverse conditions can only be satisfied if either the density n0or∇u k(r)vanish identically. Thefirst possibility is obviously ruled out,while the second contradicts our initial assumption that u k(r)is not a constant.This concludes the proof of the Runge-Gross theorem.2.3Time-dependent Kohn-Sham equationsAs mentioned in section2.1,the Runge-Gross theorem asserts that all observ-ables can be calculated with the knowledge of the one-body density.Nothing is however stated on how to calculate that valuable quantity.To circumvent the cumbersome task of solving the interacting Schr¨o dinger equation,Kohn and Sham had the idea of utilizing an auxiliary system of non-interacting (Kohn-Sham)electrons,subject to an external local potential,v KS[7].This potential is unique,by virtue of the Runge-Gross theorem applied to the non-interacting system,and is chosen such that the density of the Kohn-Sham electrons is the same as the density of the original interacting system.In the time-dependent case,these Kohn-Sham electrons obey the time-dependentTime-dependent Density Functional Theory9 Schr¨o dinger equationi ∂2+v KS(r,t) ϕi(r,t).(29)The density of the interacting system can be obtained from the time-dependentKohn-Sham orbitalsn(r,t)=occi|ϕi(r,t)|2.(30)Eq.(29),having the form of a one-particle equation,is fairly easy to solve numerically.We stress,however,that the Kohn-Sham equation is not a mean-field approximation:If we knew the exact Kohn-Sham potential,v KS,we would obtain from Eq.(29)the exact Kohn-Sham orbitals,and from these the correct density of the system.The Kohn-Sham potential is conventionally separated in the following wayv KS(r,t)=v ext(r,t)+v Hartree(r,t)+v xc(r,t).(31) Thefirst term is again the external potential.The Hartree potential accounts for the classical electrostatic interaction between the electronsv Hartree(r,t)= d3r n(r,t)δn(r,τ) n(r,t),(33) whereτstands for the Keldish pseudo-time.Inevitably,the exact expression of v xc as a functional of the density is unknown.At this point we are obliged to perform an approximation.It is im-portant to stress that this is the only fundamental approximation in TDDFT. In contrast to stationary-state DFT,where very good xc functionals exist, approximations to v xc(r,t)are still in their infancy.Thefirst and simplest of these is the adiabatic local density approximation(ALDA),reminiscent of the ubiquitous LDA.More recently,several other functionals were proposed,from which we mention the time-dependent exact-exchange(EXX)functional[13], and the attempt by Dobson,B¨u nner,and Gross[14]to construct an xc func-tional with memory.In the following section we will introduce the above mentioned functionals.10Miguel A.L.Marques and E.K.U.Gross2.4xc functionalsAdiabatic approximations There is a very simple procedure that allows the use of the plethora of existing xc functionals for ground-state DFT in the time-dependent theory.Let us assume that˜v xc[n]is an approximation to the ground-state xc density functional.We can write an adiabatic time-dependent xc potential asv adiabatic xc (r,t)=˜v xc[n](r)|n=n(t).(34)I.e.we employ the same functional form but evaluated at each time with the density n(r,t).The functional thus constructed is obviously local in time. This is,of course,a quite dramatic approximation.The functional˜v xc[n]is a ground-state property,so we expect the adiabatic approximation to work only in cases where the temporal dependence is small,i.e.,when our time-dependent system is locally close to equilibrium.Certainly this is not the case if we are studying the interaction of strong laser pulses with matter.By inserting the LDA functional in Eq.(34)we obtain the so-called adi-abatic local density approximation(ALDA)v ALDA xc (r,t)=v HEGxc(n) n=n(r,t).(35)The ALDA assumes that the xc potential at the point r,and time t is equal to the xc potential of a(static)homogeneous-electron gas(HEG)of density n(r,t).Naturally,the ALDA retains all problems already present in the LDA. Of these,we would like to emphasize the erroneous asymptotic behavior of the LDA xc potential:For neutralfinite systems,the exact xc potential decays as−1/r,whereas the LDA xc potential falls offexponentially.Note that most of the generalized-gradient approximations(GGAs),or even the newest meta-GGAs have asymptotic behaviors similar to the LDA.This problem gains particular relevance when calculating ionization yields(the ionization potential calculated with the ALDA is always too small),or in situations where the electrons are pushed to regions far away from the nuclei(e.g.,by a strong laser)and feel the incorrect tail of the potential.Despite this problem,the ALDA yields remarkably good excitation ener-gies(see sections4.2and4.3)and is probably the most used xc functional in TDDFT.Time-dependent optimized effective potential Unfortunately,when one is trying to write v xc as explicit functionals of the density,one encoun-ters some difficulties.As an alternative,the so-called orbital-dependent xc functionals were introduced several years ago.These functionals are written explicitly in terms of the Kohn-Sham orbitals,albeit remaining implicit den-sity functionals by virtue of the Runge-Gross theorem.A typical member of this family is the exact-exchange(EXX)functional.The EXX action is ob-tained by expanding A xc in powers of e2(where e is the electronic charge),Time-dependent Density Functional Theory11 and retaining the lowest order term,the exchange term.It is given by the Fock integralA EXX x=−1|r−r |.(36) From such an action functional,one seeks to determine the local Kohn-Sham potential through a series of chain rules for functional derivatives.The pro-cedure is called the optimized effective potential(OEP)or the optimized potential method(OPM)for historical reasons[15,16].The derivation of the time-dependent version of the OEP equations is very similar to the ground-state case.Due to space limitations we will not present the derivation in this article.The interested reader is advised to consult the original paper[13],one of the more recent publications[17,18],or the chapter by E.Engel contained in this volume.Thefinal form of the OEP equation that determines the EXX potential is0=occj t1−∞dt d3r [v x(r ,t )−u x j(r ,t )](37)×ϕj(r,t)ϕ∗j(r ,t )G R(r t,r t )+c.c.The kernel,G R,is defined byi G R(r t,r t )=∞k=1ϕ∗k(r,t)ϕk(r ,t )θ(t−t ),(38)and can be identified with the retarded Green’s function of the system.More-over,the expression for u x is essentially the functional derivative of the xc action in relation to the Kohn-Sham wave-functionsu x j(r,t)=1δϕj(r,t).(39)Note that the xc potential is still a local potential,albeit being obtained through the solution of an extremely non-local and non-linear integral equa-tion.In reality,the solution of Eq.(37)poses a very difficult numerical prob-lem.Fortunately,by performing an approximationfirst proposed by Krieger, Li,and Iafrate(KLI)it is possible to simplify the whole procedure,and ob-tain an semi-analytic solution of Eq.(37)[19].The KLI approximation turns out to be a very good approximation to the EXX potential.Note that both the EXX and the KLI potential have the correct−1/r asymptotic behavior for neutralfinite systems.A functional with memory There is a very common procedure for the construction of approximate xc functionals in ordinary DFT.It starts with12Miguel A.L.Marques and E.K.U.Grossthe derivation of exact properties of v xc,deemed important by physical ar-guments.Then an analytical expression for the functional is proposed,such that it satisfies those rigorous constraints.We will use this recipe to generate a time-dependent xc potential which is non-local in time,i.e.that includes the“memory”from previous times[14].A very important condition comes from Galilean invariance.Let us look at a system from the point of view of a moving reference frame whose origin is given by x(t).The density seen from this moving frame is simply the density of the reference frame,but shifted by x(t)n (r,t)=n(r−x(t),t).(40) Galilean invariance then implies[20]v xc[n ](r,t)=v xc[n](r−x(t),t).(41) It is obvious that potentials that are both local in space and in time,like the ALDA,trivially fulfill this requirement.However,when one tries to deduce an xc potential which is non-local in time,onefinds condition(41)quite difficult to satisfy.Another rigorous constraint follows from Ehrenfest’s theorem which re-lates the acceleration to the gradient of the external potentiald2dt2 d3r r n(r,t)=− d3r n(r,t)∇v ext(r).(43)In the same way we can write Ehrenfest’s theorem for the Kohn-Sham system d2Time-dependent Density Functional Theory13 i.e.the total xc force of the system is zero.This condition reflects Newton’s third law:The xc effects are only due to internal forces,the Coulomb inter-action among the electrons,and should not give rise to any net force on the system.A functional that takes into account these exact constraints can be con-structed[14].The condition(46)is simply ensured by the expression1F xc(r,t)=R(t |r,t)=j(R,t )∂tdt Πxc(n(R,t ),t−t ).(50)n(r,t)∇Finally,an expression for v xc can be obtained by direct integration of F xc (see[14]for details).2.5Numerical considerationsAs mentioned before,the solution of the time-dependent Kohn-Sham equa-tions is an initial value problem.At t=t0the system is in some initial state described by the Kohn-Sham orbitalsϕi(r,t0).In most cases the initial state will be the ground state of the system(i.e.,ϕi(r,t0)will be the solution of the ground-state Kohn-Sham equations).The main task of the computational physicist is then to propagate this initial state until somefinal time,t f.14Miguel A.L.Marques and E.K.U.GrossThe time-dependent Kohn-Sham equations can be rewritten in the inte-gral formϕi(r,t f)=ˆU(t f,t0)ϕi(r,t0),(51) where the time-evolution operator,ˆU,is defined byˆU(t ,t)=ˆT exp −i t t dτˆH KS(τ) .(52)Note thatˆH KS is explicitly time-dependent due to the Hartree and xc po-tentials.It is therefore important to retain the time-ordering propagator,ˆT, in the definition of the operatorˆU.The exponential in expression(52)is clearly too complex to be applied directly,and needs to be approximated in some suitable manner.To reduce the error in the propagation from t0 to t f,this large interval is usually split into smaller sub-intervals of length ∆t.The wave-functions are then propagated from t0→t0+∆t,then from t0+∆t→t0+2∆t and so on.The simplest approximation to(56)is a direct expansion of the exponen-tial in a power series of∆tˆU(t+∆t,t)≈kl=0 −iˆH(t+∆t/2)∆t lTime-dependent Density Functional Theory15 i)Obtain an estimate of the Kohn-Sham wave-functions at time t+∆t by propagating from time t using a“low quality”formula forˆU(t+∆t,t).The expression(53)expanded to third or forth order is well suited for this purpose. ii)With these wave-functions construct an approximation toˆH(t+∆t)and toˆU(t+∆t/2,t+∆t).iii)Apply Eq.(55).This procedure leads to a very stable propagation.The split-operator method In afirst step we neglect the time-ordering in Eq.(52),and approximate the integral in the exponent by a trapezoidal rule ˆU(t+∆t,t)≈exp −iˆH KS(t)∆t =exp −i(ˆT+ˆV KS)∆t .(56) We note that the operators exp −iˆV KS∆t and exp −iˆT∆t are diagonal respectively in real and Fourier spaces,and therefore trivial to apply in those spaces.It is possible to decompose the exponential(56)into a form involving only these two operators.The two lowest order decompositions are exp −i(ˆT+ˆV KS)∆t =exp −iˆT∆t exp −iˆV KS∆t +O(∆t2)(57) andexp −i(ˆT+ˆV KS)∆t =exp −iˆT∆t2+O(∆t3).(58) For example,to apply the splitting(58)toϕ(r,t)we start by Fourier trans-forming the wave-function to Fourier space.We then apply exp −iˆT∆t16Miguel A.L.Marques and E.K.U.GrossInstead perturbation theory may prove sufficient to determine the behavior of the system.We will focus on the linear change of the density,that allows us to calculate,e.g.,the optical absorption spectrum.Let us assume that for t<t0the time-dependent potential v TD is zero–i.e.the system is subject only to the nuclear potential,v(0)–and furthermore that the system is in its ground-state with ground-state density n(0).At t0 we turn on the perturbation,v(1),so that the total external potential now consists of v ext=v(0)+v(1).Clearly v(1)will induce a change in the density. If the perturbing potential is sufficiently well-behaved(like almost always in physics),we can expand the density in a perturbative seriesn(r,t)=n(0)(r)+n(1)(r,t)+n(2)(r,t)+ (60)where n(1)is the component of n(r,t)that depends linearly on v(1),n(2) depends quadratically,etc.As the perturbation is weak,we will only be con-cerned with the linear term,n(1).In frequency space it readsn(1)(r,ω)= d3r χ(r,r ,ω)v(1)(r ,ω).(61)The quantityχis the linear density-density response function of the sys-tem.In other branches of physics it has other names,e.g.,in the context of many-body perturbation theory it is called the reducible polarization func-tion.Unfortunately,the evaluation ofχthrough perturbation theory is a very demanding task.We can,however,make use of TDDFT to simplify this process.We recall that in the time-dependent Kohn-Sham framework,the density of the interacting system of electrons is obtained from afictitious system of non-interacting electrons.Clearly,we can also calculate the linear change of density using the Kohn-Sham systemn(1)(r,ω)= d3r χKS(r,r ,ω)v(1)KS(r ,ω).(62)Note that the response function that enters Eq.(62),χKS,is the density re-sponse function of a system of non-interacting electrons and is,consequently, much easier to calculate than the full interactingχ.In terms of the unper-turbed stationary Kohn-Sham orbitals it readsχKS(r,r ,ω)=limη→0+∞jk(f k−f j)ϕj(r)ϕ∗j(r )ϕk(r )ϕ∗k(r)Time-dependent Density Functional Theory17 v(1)KS.This latter quantity can be calculated explicitly from the definition of the Kohn-Sham potentialv(1)KS(r,t)=v(1)(r,t)+v(1)Hartree(r,t)+v(1)xc(r,t).(64) The variation of the external potential is simply v(1),while the change in the Hartree potential isv(1)(r,t)= d3r n(1)(r ,t)Hartreen(1)(r ,t ).(66)δn(r ,t )It is useful to introduce the exchange-correlation kernel,f xc,defined byδv xc(r,t)f xc(r t,r t )=n(1)(r ,ω).|x−r |+f xc(x,r ,ω)From Eq.(61)and Eq.(68)trivially follows the relationχ(r,r ,ω)=χKS(r,r ,ω)+(69) d3x d3x χ(r,x,ω) 118Miguel A.L.Marques and E.K.U.Gross3.2The xc kernelAs we have seen in the previous section,the main ingredient in linear response theory is the xc kernel.f xc,as expected,is a very complex quantity that includes–or,in other words,hides–all non-trivial many-body effects.Many approximate xc kernels have been proposed in the literature over the past years.The most ancient,and certainly the simplest is the ALDA kernelf ALDA xc (r t,r t )=δ(r−r )δ(t−t )f HEGxc(n) n=n(r,t),(70)wheref HEG xc (n)=dn(r,t)[u x k(r,t)+c.c].(72)Using the definition(67)and after some algebra we arrive at thefinal form of the PGG kernelf PGG x (r t,r t )=−δ(t−t )1|r−r || occ kϕk(r)ϕ∗k(r )|2dn2 n HEG xc(n) ≡f0(n),(74)Time-dependent Density Functional Theory19where HEGxc ,the xc energy per particle of the homogeneous electron gas,isknown exactly from Monte-Carlo calculations[23].Also the infinite frequency limit can be written as a simple expressionlim q→0f HEGxc(q,ω=∞)=−43dn2/3 +6n1dnHEG xc(n)πf HEGxc(q,ω)πf HEGxc(q,ω)−f HEGxc(q,∞)15ω3/2.(79)The real part can be obtained with the help of the Kramers-Kronig relationslim ω→∞ f HEGxc(q=0,ω)=f∞(n)+23π(1+β(n)ω2)520Miguel A.L.Marques and E.K.U.GrossFig.1.Real and imaginary part of the parametrization for f HEGxc.Figure reproduced from Ref.[25].The coefficientsαandβare functions of the density,and can be determined uniquely by the zero and high frequency limits.A simple calculation yieldsα(n)=−A[f∞(n)−f0(n)]53,(83) where A,B>0and independent of n.Once again,by applying the Kramers-Kronig relations we can obtain the corresponding real part of f HEGxcf HEGxc (q=0,ω)=f∞+2√π√√2Π 1−r√2Π 1+r√1+βω2and E andΠare the elliptic integrals of second andthird kind.In Fig.1we plot the real and imaginary part of f HEGxc for twodifferent densities(r s=2and r s=4,where r s is the Wigner-Seitz radius, 1/n=4πr3s/3).The ALDA corresponds to approximating these curves by their zero frequency value.For very low frequencies,the ALDA is naturally a good approximation,but at higher frequencies it completely fails to reproducethe behavior of f HEGxc .To understand how the ALDA can yield such good excitation energies, albeit exhibiting such a mediocre frequency dependence,we will look at a spe-cific example,the process of photo-absorption by an atom.At low excitation frequencies,we expect the ALDA to work.As we increase the laser frequency, we start exciting deeper levels,promoting electrons from the inner shells of the atom to unoccupied states.The atomic density increases monotonically as we approach the nucleus.f xc corresponding to that larger density(lower。