text:
#set( $symbol_pound = '#' )
#set( $symbol_dollar = '$' )
#set( $symbol_escape = '\' )

package integration.tests;

import dom.todo.ToDoItem;
import dom.todo.ToDoItemContributions;
import dom.todo.ToDoItems;
import fixture.todo.scenarios.ToDoItemsRecreateAndCompleteSeveral;

import java.util.List;
import javax.inject.Inject;
import org.junit.Before;
import org.junit.Test;
import org.apache.isis.applib.fixturescripts.FixtureScripts;

import static org.hamcrest.CoreMatchers.containsString;
import static org.hamcrest.CoreMatchers.is;
import static org.hamcrest.CoreMatchers.not;
import static org.hamcrest.CoreMatchers.nullValue;
import static org.junit.Assert.assertThat;

public abstract class ToDoItemContributionsIntegTest extends AbstractToDoIntegTest {

    ToDoItemsRecreateAndCompleteSeveral fixtureScript;

    @Before
    public void setUpData() throws Exception {
        fixtureScript = new ToDoItemsRecreateAndCompleteSeveral();
        fixtureScripts.runFixtureScript(fixtureScript, null);
    }

    @Inject
    FixtureScripts fixtureScripts;
    @Inject
    ToDoItems toDoItems;
    @Inject
    ToDoItemContributions toDoItemContributions;

    ToDoItemContributions toDoItemContributionsWrapped;
    ToDoItem toDoItem;

    @Before
    public void setUp() throws Exception {

        toDoItem = wrap(fixtureScript.lookup("to-do-items-recreate-and-complete-several/to-do-item-complete-for-buy-stamps/item-1", ToDoItem.class));
        assertThat(toDoItem, is(not(nullValue())));

        toDoItemContributionsWrapped = wrap(toDoItemContributions);
    }

    public static class Actions {
        public static class UpdateCategory extends ToDoItemContributionsIntegTest {

            @Test
            public void happyCase() throws Exception {

                // when
                toDoItemContributionsWrapped.updateCategory(toDoItem, ToDoItem.Category.Professional, ToDoItem.Subcategory.Consulting);

                // then
                assertThat(toDoItem.getCategory(), is(ToDoItem.Category.Professional));
                assertThat(toDoItem.getSubcategory(), is(ToDoItem.Subcategory.Consulting));

                // when
                toDoItemContributionsWrapped.updateCategory(toDoItem, ToDoItem.Category.Domestic, ToDoItem.Subcategory.Chores);

                // then
                assertThat(toDoItem.getCategory(), is(ToDoItem.Category.Domestic));
                assertThat(toDoItem.getSubcategory(), is(ToDoItem.Subcategory.Chores));
            }

            @Test
            public void categoryCannotBeNull() throws Exception {

                // when, then
                expectedExceptions.expectMessage("'Category' is mandatory");
                toDoItemContributionsWrapped.updateCategory(toDoItem, null, ToDoItem.Subcategory.Chores);
            }

            @Test
            public void subcategoryCanBeNull() throws Exception {

                // when, then
                toDoItemContributionsWrapped.updateCategory(toDoItem, ToDoItem.Category.Professional, null);
            }

            @Test
            public void subcategoryMustBelongToCategory() throws Exception {

                // when, then
                expectedExceptions.expectMessage(containsString("Invalid subcategory"));
                toDoItemContributionsWrapped.updateCategory(toDoItem, ToDoItem.Category.Professional, ToDoItem.Subcategory.Chores);
            }
        }

        public static class SimilarTo extends ToDoItemContributionsIntegTest {

            @Test
            public void happyCase() throws Exception {

                // when
                List<ToDoItem> similarItems = toDoItemContributionsWrapped.similarTo(toDoItem);

                // then
                assertThat(similarItems.size(), is(6));
            }

        }
    }

    public static class Properties {
        public static class Priority extends ToDoItemContributionsIntegTest {

            private List<ToDoItem> notYetComplete;

            @Before
            public void setUp() throws Exception {
                notYetComplete = wrap(toDoItems).notYetComplete();
            }

            @Test
            public void happyCase() throws Exception {
                assertPriority(0, 1);
                assertPriority(1, 2);
                assertPriority(2, 4);
                assertPriority(3, 6);
                assertPriority(4, 5);
                assertPriority(5, 7);
                assertPriority(6, 9);
                assertPriority(7, 8);
                assertPriority(8, 3);
                assertPriority(9, 10);
            }

            private void assertPriority(final int n, final int priority) {
                assertThat(toDoItemContributions.relativePriority(notYetComplete.get(n)), is(Integer.valueOf(priority)));
            }
        }
    }

}

redpajama_set_name: RedPajamaGithub


Last weekend, I finally followed the recommendations from our office's annual physical exam. I got my first set of maintenance meds.
Gargh! Cholesterol. Yep, definitely #thatmoment when you know that the saying "Health is wealth" is so true.
My wallet felt lighter after that moment. Well, on the upside I got a Mercury Drug suki card for free. So I guess that is a plus.
But seriously, at 45goingplaces 😊, health has become a priority for me.
I wanna be that lola on the podium of backwater weekend races, doing wicked yoga early in the morning and HIITing it hard regularly.
Today, I commit to complete 8 weeks of on-plan workout and meals.
I know this will get exhausting. It will get tiring, stale. Or my backpocket excuse, life and work will get in the way.
But today, I realised if I wanna shake off this maintenance meds business I need to commit and sustain what I've started.
As long as I keep trying, I'll never really lose. It's only when you give up that you will truly lose. I resolve not beat myself up for that donut or a missed workout. Instead, I will pick up where I left off, not put it off any longer.
Yep, #thatmoment when you're beat up from a hard workout but excited to push on because the gains are bigger than how tired and exhausted you feel at the moment. A healthy future ahead is worth the bucket of sweat and … I really truly wanna be that crazy lola racing 5kms.
Have you had a similar moment?
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Tag @fitnessfirst_ph and use #ThatMoment #FitnessFirstPH #FFbranchname (e.g. #FFPlatinumAura) and get a chance to win Php10,000 worth of exciting prizes from Fitness First! Your story will also be featured in Fitness First Social media pages.
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This necklace with birthstone and hand stamped names are perfect for mothers, grandma or special someone. This necklace comes with sterling silver one 5/8" 20g discs. One channel drop birthstones and beaded chain also come with this necklace. You can choose more disc from drop down above. All comes with birthstone of your choice.
1. Personalization for hand stamped disc (front side only)e.g. Disc 1: Mary, Disc 2: John etc.


Dianne is doing so well so fast that they were able to go home to Bellingham yesterday! YAY! Based on observing how much work it is for Dianne and Vince to do all the medical processes required right now we (Debbie and I) think that they could use some help in Bellingham.
As many of you know, Dianne and I are in Seattle seeking treatment at the UW Medical Center. We have created this Care Bridge web page to bring you up to date on what's going on and as a vehicle for sharing information going forward. As you might imagine this is a very busy and emotional time for our family. We are very grateful for your concern and friendship.
On and off for about a year now, Dianne has been dealing with a skin problem. Unfortunately, the problem masked early symptoms that are also signs of leukemia. There are plenty of details that are often part of these situations, at this point they aren't relevant to what needs to happen moving forward or to any help that you all might give us.
Dianne has acute myeloid leukemia, AML, a type of leukemia that doesn't manifest in the blood tests but interferes with blood cell production in the bone marrow.
The bad news is that it is that AML is a really bad form of leukemia. The good news is that The Cancer Care Alliance in Seattle is one of the leading centers for the research and treatment of this disease in the world.
The medical staff is completing more in-depth, genetic tests of her bone marrow to see what treatment protocols will work best for her particular version of AML. We are being presented with options for different levels of treatment, which have varying degrees of risk versus benefit. With as much information as possible, we will decide which next steps we (she) wants to take. But in general, the next steps are weeks of chemo therapy, of varying degrees of harshness, in order to bring Dianne's AML into remission. If that is successful we understand that remission generally lasts for a year or so, with the main statistical probability that the cancer will return, and she will need a stem cell transplant. Those are the very basics at this time.
The nursing unit phone number is (206) 598-0700.
Dianne's direct room phone number is (206) 906 7492.
Responses to questions people have asked, will ask, or want to ask.
· Dianne's Care Bridge web page will be our vehicle for communicating consistent and timely information to you so that Vince can focus on Dianne and the doctors. We'd like to keep his cell phone free for critical family communications.
· Please feel free to leave comments and thoughts on the site. We want to stay connected with all of you.
o How long the treatment is going to be?
o Whether we will be in-patient or outpatient?
o What all the physical effects from the chemotherapy will be (though they are the usual that intense chemo cancer treatment impacts).
· People are welcome to visit if they are not ill with colds, flu or any other infectious disease. This really is standard for any hospital but is particularly important for cancer units.
· Short visits are best. Dianne tires easily and is having a lot of procedures and assessments by doctors.
1. Check here the day before in case there is a reason the following day is not good—we will post information about that.
3. Don't always expect to see Vince—he may not be there or may be involved with other issues.
Visiting Status Today July 10—Dianne is going to be in and out of tests—but this is a good day to visit.
For those coming from out of town, use Google Maps to find the medical center, which is on the south end of the UWcampus. But definitely call ahead. We wouldn't want you make that long a trip and not being able to get to see her.
and then check into the nursing station right there.
And remember to wash and gel your hands. Can't do too much of that.
o You are welcome to donate blood. It will go into the common pool but I think it might help if you are saying that you are donating for a certain person. I have read that blood of younger people is better than that of older, but I am not sure if this is a confirmed, hard truth. And certainly having blood available in the blood banks is better than not having it available, and it does get old and less useful the longer is sits after being donated.
o Actually, hospital food is really good these days, especially in such a fancy hospital as the U of W. We don't have anywhere to store the food in Dianne's room, and we can't take back out to others. So bringing food, though natural and intuitively spontaneous thing people do, is not particularly helpful right now for Dianne.
- Unfortunately, fresh flowers or plants of any kind cannot be taken into the unit.
This information can and may change. We will try to keep things updated on this web page, but please don't assume that it all is. And please be patient with us if updates are not forthcoming. We don't know what kind of side effects Dianne might experience from her treatment. Plans may change,and we may need to go silent for a while as we adjust to new outcomes and alter our decisions about how to go forward.


text:
North Island Copper and Gold Project, Vancouver Island
Producer of
Copper-gold, and molybdenum concentrate
Mine Type
Open-pit
Vancouver Island, Canada
Mine Life
Annual Production
82Mlbs of copper, 79,000oz of gold, and 3Mlbs of molybdenum
Owner/Developer
NorthIsle Copper and Gold
The North Island copper-gold project is located in the north end of Vancouver Island, British Columbia, Canada. Image courtesy of Guanaco.
Diorite is one of the major lithological units found at Hushamu pit of the North Island project. Image courtesy of Rygel, M.C.
Access to the mine from Port Hardy is by paved roads and well-maintained logging roads. Image courtesy of The Interior.
Located in Vancouver Island, British Columbia, Canada, the North Island copper and gold project is being developed by NorthIsle Copper and Gold through its fully owned subsidiary North Island Mining.
The preliminary economic assessment (PEA) study for the project was completed in September 2017, which indicated that the annual production from the Canadian copper-gold project is estimated to be 82Mlb of copper, 79,000oz of gold, and 3Mlb of molybdenum.
The mine is estimated to have a life of 22 years and an overall production of 1.8 billion pounds of copper, 1.7Moz of gold, and 55Mlb of molybdenum.
Geology and mineralisation of North Island project
The copper-gold project is a combination of 212 mixed legacy and cell mineral claims covering 33,447ha. It comprises two main deposits, namely Hushamu and Red Dog, along with five partially explored copper-gold porphyry deposits.
The dominant rocks of the Hushamu deposit are believed to be from the early to mid-Jurassic Bonanza Group volcanics and the Mid-Jurassic Island Plutonic Suite.
"The copper-gold project is a combination of 212 mixed legacy and cell mineral claims covering 33,447ha."
Major lithological units found at Hushamu are andesite, diorite, quartz-feldspar porphyry, hydrothermal breccia, and late breccia. The massive andesite is host to most of the alteration and mineralisation found at the deposit.
The deposit hosts four main alteration styles namely silica-clay-pyrite (SCP), silica-clay-zunyite (SCZ), chlorite-magnetite (CMG), and propylitic. The dominant structures are northwest and northeast normal and strike-slip faults, while the dominant fault is called the Hushamu Fault, located in the main valley at the north side of the deposit.
The deposit mineralisation was mainly intruded by Rose Porphyry, a granite porphyry.
The deposit is contained within the west-northwest trending quartz-magnetite breccia, held in altered Bonanza Group rocks adjacent to quartz-feldspar porphyritic dykes.
North Island project reserves
The Hushamu deposit is estimated to contain indicated resources of 1.4 billion pounds of copper, 2.8Moz of gold, and 65.7Mlb of molybdenum.
Mining at North Island
Conventional truck and shovel method of mining will be applied at the Hushamu and Red Dog open-pit deposits of the North Island project. Mining is expected to be conducted at a rate of 64Mtpa, producing an average of 54Mtpa over the first 12 years.
The Hushamu open-pit will be developed in four phases and is expected to extract 50,000t of ore a day, while the Red Dog deposit is estimated to extract 25,000t of ore a day.
Overland conveyors will be used to transport ore from the two deposits to the processing facilities. The mine will operate four production drills, three hydraulic shovels and two wheel loaders, and 18-220t class unit trucks.
Waste rock from Hushamu deposit will be delivered to the tailings dam construction and within the mine waste storage facility.
Ore processing
The processing plant at North Island project will be a conventional flotation plant with a nameplate capacity of 75,000tpd. It will produce separate copper/gold, molybdenum, and pyrite concentrates as saleable products.
Run-of-mine (ROM) ore from both the Hushamu and Red Dog deposits will be crushed and stockpiled before conveying to the process plant. The ore will pass through conventional semi-autogenous (SAG)/ball mill grinding for processing to a size suitable for flotation.
Copper-molybdenum concentrate will be initially produced from the flotation process, followed by the separation of molybdenum from copper in the flotation circuit. A final copper/gold concentrate and a molybdenum concentrate will be produced after separation.
Pyrite will be recovered from the bulk copper-molybdenum flotation rougher tails in a pyrite scavenger flotation circuit. The concentrate will be combined with the tails from the bulk cleaner circuit and processed in a pyrite upgrade flotation step to produce a clean saleable pyrite concentrate.
The final copper/gold and pyrite flotation concentrates will be filtered and directly loaded onto highway haul trucks for transport to the market, while the molybdenum concentrate will be filtered, dried and packaged for transport to the market.
Power supply for the mine will be made through a new 36km-long, 138kV power line connecting to an existing BC Hydro electrical substation in Port Hardy. A 28km-long, 34.5kV powerline will be constructed for power distribution across the site.
The project is accessible from Port Hardy through paved roads and well-maintained logging roads.
Contractors involved
M3 Engineering & Technology prepared the preliminary economic assessment (PEA) for the copper-gold project.
GeoMinEx prepared the geology, exploration and environmental study, while Base Metallurgical Laboratories prepared the mineral processing and metallurgical testing models.
Phil Burt Consulting Services prepared the mineral resource estimates, while Golder Associates designed the mine waste storage facility.
EMG Mining Consultants estimated the pricing for the mobile equipment to be used at the project.
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redpajama_set_name: RedPajamaCommonCrawl


Talking about self-doubt and imposter syndrome is something I think Entrepreneurs need to discuss more often.
It's tough, because you want to believe in your dream and to assure people that you know what you're doing. You should feel confident in your business pursuits. After all, you have to believe in your dream even when no one else is.
Self-doubt and imposter syndrome is a real emotional and psychological challenge that plagues more Entrepreneurs than people like to admit.
I had (and still have) my moments where I feel like a total fraud in a room with other professionals who I look up to and admire. The crazy part is that the people I look up to also suffer from imposter syndrome from time to time.
Imposter Syndrome affects everyone and a recent study done by American and German scientists revealed that, "Men with impostor syndrome tended to crumble under pressure much more than the women who suffered from the same feelings."
How did the women do?
The researchers found, "In contrast, female students with high imposter feelings responded to harsh feedback by increasing their effort and showing superior performance."
Regardless of where you are in your entrepreneurial journey, learning how to cope with self-doubt and imposter syndrome is an important skill. Below, I put together a list of how I've dealt with these negative feelings since starting my business, LN Creative Group, a couple years ago. I hope it helps you as you work through these emotions.
First thing to do is to acknowledge your feelings of self-doubt and imposter syndrome. Just notice them without engaging it. Be aware of your feelings, and then slowly figure out why you're feeling that way.
Look at your situation from another perspective and reframe your thinking. Are you putting too much pressure on yourself? Are you trying to be perfect? Your mind can race in intense situations or challenging environments. Learning how to control your mind and automatic negative thoughts is an important skill for success.
I used to respond to all challenges with a voice in my head saying, "You need to fix this! This is an emergency!" I would then get all worked up, and let my mind race. "If you don't resolve this, then you're not as good as others." It may seem silly to some (including me now), but in the heat of the moment that's how these thoughts would build in me.
Instead, I had to learn to be calm, breathe, and reframe my mindset to any challenges that comes into my life. Now, my auto reaction is to not have a reaction. I've trained myself to be my own friend and talk myself down whenever I'm feeling anxious.
There is nothing that helps me overcome my sense of self-doubt and imposter syndrome more than sharing it with my husband, Chris, or my most trusted friends. They truly help me put things into perspective and remind me of why I'm doing what I'm doing.
Sharing your experiences is an excellent method for gaining greater clarity and realizing that the world is so much bigger than you and your problems alone. It's part of the reason why I started reading about stoic philosophy to help me cope as an entrepreneur.
It's easy to go online and see someone else achieving all their goals with shiny new photos and opportunities. You're happy for them, but you get a tinge of jealously. You then start to compare yourself to them. Your mind begins to spiral and now you no longer believe how great you truly are.
Comparing yourself to someone else is a dangerous habit. The reality is... No one else is you, and that is your power.
Some of the best entrepreneurs I know keep a journal of some kind. I've kept a journal for the majority of my life not knowing what it really meant. I loved writing as a child and still do. Maybe you don't like writing, but I'm still encouraging you to write it down. Don't look at it as a chore but more of an avenue to express yourself - like painting, drawing, making music, etc.
Making the effort to mentally unload in a journal is one of the most therapeutic activities I know of. It not only helps you work through your problems, but it's really nice to simply write down all your thoughts, fears, worries in a safe place.
Have you ever felt like everyone else has it all figured out? They seem to be doing really well and everyone admires them. Truth is - nobody has any idea what they're doing. They're all just doing their best. Startups fail all the time. People change careers, attitudes, perspectives, locations all the time.
To get over self-doubt and imposter syndrome, you must realize that no one knows what they're doing. And that takes a lot of pressure off. There are no rules in life or business. You just have to figure out what works for you individually.
Everyone - and I mean everyone - who embarks on an entrepreneurial journey will come face-to-face with imposter syndrome and self-doubt at some point.
Try out some of these tips and let me know where you are on your entrepreneurial journey. Leave me a link to your website in the comments section and how you deal with imposter syndrome below.


text:
Falling for You 2018 While planning a fundraising event, a small New England town's radio station manager meets her match in a visiting businessman who can't seem to see beyond the screen of his laptop computer, until she ropes him into participating in her bachelor bake-off to help save the station.
Director: Peter DeLuise
Actors: Agape Mngomezulu, Alison Wandzura, Chris Shields, Christine Chatelain, Lini Evans, Taylor Cole, Tyler Hynes
From the Land of the Moon 2016
From the Land of the Moon 2016 An adaptation of Milena Agus' eponymous novel set after WWII, "Mal de Pierres" ("Mal di Pietre") spans 20 years, following the destiny of…
Country: Belgium, Canada, France
Tera Intezaar 2017 Tera Intezaar (English: I'm waiting for you) is an upcoming 2017 Bollywood musical romantic film, directed by Raajeev Walia and produced by Aman Mehta and Bijal Mehta.
Genre: Romance, Thriller
Falling Inn Love 2019 When a San Francisco exec wins a New Zealand inn, she ditches city life to remodel and flip the rustic property with help from a handsome…
Rocky Mountain Christmas 2017 Sarah Davis heads to her uncle's struggling ranch to escape New York and the spotlight from a recent breakup. Returning home for the first time since…
A Stroke of Luck 2016
A Stroke of Luck 2016 Villaviciosa de al Lado is a peaceful town stuck in time, living exclusively of the fame of a 200 years-old spa. Therefore, city hall's debts…
Time Freak 2018 Stillman, a heartbroken physics student, builds a time machine when his girlfriend breaks up with him. Going back in time, he attempts to save their relationship by…
Stock Option 2015
Stock Option 2015 A down-on-his-luck stock broker becomes homeless and takes refuge in front of an office building where a famous print model works. After he heroically rescues her from…
Sparks & Embers 2015
Sparks & Embers 2015 Tom and Eloise meet in inauspicious circumstances, trapped in a stuck lift after Eloise has Tom fired. Nevertheless, sparks fly. 5 years later, they have broken…
A Novel Romance 2015
A Novel Romance 2015 Romance novelist Liam Bradley (Dylan Bruce) has already found massive success with three books written under the pen name Gabriel August, but he's mysteriously unknown to…
Genre: Comedy, Drama, Romance, TV Movie
Frantz 2016
Frantz 2016 In the aftermath of WWI, a young German who grieves the death of her fiancé in France meets a mysterious French man who visits the fiance's grave to…
Country: France, Germany
The Bookshop 2018 Set in an small English town in 1959, the story of a woman who decides, against polite but ruthless local opposition, to open a bookshop, a decision…
Country: Germany, Spain, UK
Wedding Unplanned
Wedding Unplanned 2017 When she discovers a wedding planer's business card, Alexia instantly says, "YES" to Mathias unaware that it belongs to his mistress. The groom is now trapped between…
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text:
\section{Introduction}
Perovskite manganites, $R$MnO$_3$ ($R^{3+}$ = rare earth cation), show a great variety of structural, magnetic and electronic phases whose coexistence and interplay give rise to the large diversity of their physical properties. Orthorhombic $R$MnO$_3$ (o-$R$MnO$_3$) exhibiting frustrated magnetic orderings are of particular interest as they belong to the family of so-called magnetoelectric multiferroics - materials, where magnetic and ferroelectric orders are simultaneously presented \cite{{Hill:2000fx},{Spaldin:2005jx},{Cheong:2007uw}}. Indeed, it has been shown experimentally, that the establishment of a spiral ordering of Mn$^{3+}$ spins in TbMnO$_3$ and DyMnO$_3$ is accompanied by the appearance of a spontaneous electric polarization which can be manipulated by an applied magnetic field \cite{Kimura:2003wn}. Recently a magnetically induced electric polarization was also observed in o-HoMnO$_3$ \cite{Lorenz:2007bq}, which has an E-type antiferromagnetic order (E-AFM) \cite{PhysRev.100.545}. Despite the fact that these effects occur at quite low temperatures, the understanding of their mechanisms is important for the fundamental physics of magnetoelectric phenomena and for potential development of multifunctional devices.

In this work we address the question of the origin of the frustrated magnetic orderings which cause the multiferroic properties in o-$R$MnO$_3$. According to experiment, the magnetic structure in the series of o-$R$MnO$_3$ evolves from A-AFM to the spiral and then to the E-AFM state with decreasing radius of the $R$ cation, which favors the enhancement of orthorhombic distortion. This in turn changes the relative strength of nearest-neighbor (NN) and further neighbor exchange interactions between Mn spins in these materials \cite{{Zhou:2006hj},{Kimura:2003kq}}. This evolution of the magnetic order is usually described within the framework of a Heisenberg model with competing NN and next-nearest-neighbor (NNN) exchanges. Indeed, qualitatively, this model gives the spiral as a ground state for a certain ratio between NN and NNN couplings \cite{{Cheong:2007uw}, {Mochizuki:2009br}, {Kaplan:2009fx}}. However, as we will show in details in Sec.\ \ref{sec:2b}, application of this model for quantitative description of the exchanges in o-$R$MnO$_3$ gives contradictory results. Moreover, it was shown recently, that the E-AFM state cannot  be obtained from the Heisenberg Hamiltonian \cite{Kaplan:2009fx, Kaplan:pRvO8IfS}.
\begin{figure}[h]
\centering
{\includegraphics[scale=0.46, trim=0cm 0cm 0cm 0cm]{structureall1.eps}}
\caption
{(Color online) Crystal structure of o-$R$MnO$_3$: (a) - side view; (b) - top view ($R$ ions are not shown). (c) and (d) represent the normal modes of Jahn-Teller distortion $Q_2$ and $Q_3$, respectively.}
\label{fig1}
\end{figure}

Here we present the results of our studies of the microscopic magnetic couplings in the series of o-$R$MnO$_3$ using first-principles electronic structure calculations with the goal of finding a model Hamiltonian (excluding effects due to spin-orbit coupling) which can accurately describe the magnetism in these materials. First we consider TbMnO$_3$ with spiral spin ordering. We map the exchanges in this compound onto the Heisenberg model and find a clear deviation from Heisenberg-like behavior. We investigate the extent to which this deviation originates from the presence of the orbital ordering in TbMnO$_3$ and show that the coupling between magnetic and orbital degrees of freedom cannot provide the observed deviation. Then we explore the effect of exchange couplings of higher orders than the bilinear exchange (biquadratic and four-spin ring couplings), which are usually neglected. We demonstrate that the higher order contributions are significant in TbMnO$_3$ and other o-$R$MnO$_3$ with small radii of the $R$ cation and they have to be included in the model Hamiltonian for an accurate description of the magnetic properties of orthorhombic manganites.

This article is organized as follows: in Sec.\ \ref{sec2} we describe the crystal structure, the orbital ordering and its relation to the magnetic properties in o-$R$MnO$_3$, and explain the motivation of our research. Here we also introduce the methods which we use in our calculations and specify the computational details. In Sec.\ \ref{sec3} we calculate the microscopic exchange couplings in TbMnO$_3$ and show that they cannot be described by the Heisenberg Hamiltonian. In Sec.\ \ref{sec4} we discuss the possible sources of the non-Heisenberg behavior in TbMnO$_3$; in particular, we investigate the effects of orbital ordering, structural distortions and higher order exchange couplings. In Sec.\ \ref{sec5} we extend our analysis on the other representatives of the o-$R$MnO$_3$ series, namely, PrMnO$_3$ and LuMnO$_3$. Finally, in Sec.\ \ref{sec6} we summarize our work and give a conclusion.

\section{Motivation, theoretical background and methods}
\label{sec2}
\subsection{Jahn-Teller and GdFeO$_3$-type distortions in o-$R$MnO$_3$}
\label{sec2a}
The o-$R$MnO$_3$ have an orthorhombically distorted perovskite structure (see Fig.\ \ref{fig1}) with space group \textit{Pbnm} ($\#$62) and 20 atoms per unit cell \cite{{Kimura:2003wn},{Alonso:2000kv},{Ederer:2007fc},{Kovacik:2010bf}}. The deviation from the perfect cubic perovskite structure includes the Jahn-Teller distortion of the MnO$_6$ octahedra \cite{Kanamori:1960ki}, their cooperative tiltings \cite{Woodward:1997ci} (the so-called GdFeO$_3$-type, GFO, distortion) and small antiferroelectric displacements of $R$ cations from their ideal positions \cite{Benedek:2013jl}. While the latter structural distortion has been shown to influence the ferroelectric properties, it's  effect on the magnetism is negligible and we do not consider it in this work.

In o-$R$MnO$_3$ each Mn$^{3+}$ ion resides in the middle of an oxygen octahedron with four electrons in $3d$ levels. The crystal field of the perfect octahedron splits the fivefold degenerate $d$ levels into triply-degenerate $t_{2g}$ lower-energy levels and doubly-degenerate $e_g$ levels with higher energy. Electrons occupy the orbitals according to Hund's rules and the Pauli principle, which leads to full occupation of the spin majority $t_{2g}$ states and single occupation of the spin majority $e_g$ states.  Lowering of the symmetry of the crystal field due to the Jahn-Teller effect \cite{Jahn:1937gp} lifts the degeneracy of the $e_g$ electronic state and favors the occupation of a certain orbital which can be represented as a superposition of $d_{z^2}$ and $d_{x^2-y^2}$-states \cite{Kugel:1973ub}:
\begin{equation}
|\psi\rangle=\cos\left(\frac{\theta}{2}\right)|d_{z^2}\rangle+\sin\left(\frac{\theta}{2}\right)|d_{x^2-y^2}\rangle
\label{orbmixing}
\end{equation}
The state $|\psi \rangle$ is uniquely defined by the angle $\theta$ which is called the orbital mixing angle. The corresponding distortion of the octahedron can be written as a linear combination of two normal Jahn-Teller modes $Q_2$ and $Q_3$ \cite{{VanVleck:1939hw},{Kanamori:1960ki}} (Fig.\ \ref{fig1} (c) and (d), respectively):
\begin{equation}
Q=Q_3\cos\varphi+Q_2\sin\varphi
\end{equation}
The value of $\varphi$ can be estimated with the simple formula
\begin{equation}
\varphi=\arctan\left(\frac{Q_2}{Q_3}\right)=\arctan\left(\frac{\sqrt{3}(l-s)}{2m-l-s}\right),
\label{eq5}
\end{equation}
where  $l$, $m$ and $s$ are the lengths of the long, medium and short Mn-O bonds in the octahedron \cite{Khomskii:2005ws}. The ground state value of $\theta$ is determined by the balance between the energy gain due to the orbital-lattice interaction and the elastic energy cost  \cite{Brink:2004gp}. For a single octahedron this occurs at $\varphi=\theta$.

Since the oxygen octahedra are interconnected, their distortions and, therefore, the occupied orbital states on neighboring Mn ions are not independent. Below a certain temperature, this leads to a long-range orbital ordering with the orbital mixing angles for two neighboring Mn sites $i$ and $j$ in the $ab$ plane related by: $\theta_i=-\theta_j$ (antiferro-orbital orientation). For nearest neighbors along the $c$ direction they are equal (ferro-orbital orientation).

The GFO distortion is characterized by almost rigid cooperative rotations of the MnO$_6$-octahedra, which result in the reduction of Mn-O-Mn bond angles and  O(1)-O(2) distances (see Fig.\ \ref{fig1}(b)). In the series of o-$R$MnO$_3$ this distortion increases with decreasing radius of the $R$ cation from La to Lu.
\subsection{\label{sec:2b} Frustrated magnetism in o-$R$MnO$_3$}
The combination of Jahn-Teller and GFO distortions in o-$R$MnO$_3$ determines their magnetic properties \cite{Lee:2013jl}. According to the Goodenough-Kanamori-Anderson rules \cite{{Goodenoughbook},{Goodenough:1955tg},{Kanamori:1959wa}}, the presence of the orbital ordering of the type which was described in the previous section favors FM exchange coupling between neighboring Mn spins in the $ab$ planes and AFM coupling along the crystallographic $c$ direction. This promotes the establishment of A-AFM ordering  for $R$=La...Gd. However, further decreasing the size of $R$-cation in the series of o-$R$MnO$_3$ (and, therefore, increasing the GFO distortion) causes the transition to the spiral ($R$=Tb, Dy) and then E-AFM states ($R$=Ho...Lu). One can consider the change in the relative strength of FM NN and AFM NNN couplings in the $ab$ plane as an origin of this transition. Indeed, increasing GFO distortion decreases NN exchange as it strongly depends on the Mn-O-Mn bond angles. On the other hand, it enhances the AFM exchange between NNNs along the $b$ axis through the path Mn-O(1)-O(2)-Mn due to the reduction of O(1)-O(2) distances (see Fig.\ \ref{fig1} (b)). This strong AFM NNN exchange causes magnetic frustration \cite{Kimura:2003kq}.

The simplest microscopic model which is often used to discuss this evolution of the magnetic order is the Heisenberg model:
\begin{equation}
H_\mathrm{Heis}=\sum_{\langle i,j\rangle}{J_{ij}\mathbf{S_i}\cdot \mathbf{S_j},}
\label{eqHeis}
\end{equation}
where $J_{ij}$ indicates exchange interactions between spins $\mathbf{S_i}$ and $\mathbf{S_j}$. As exchange interactions are short-ranged, usually only the couplings between first- and second-nearest neighbors are taken into account \cite{Cheong:2007uw}. This model qualitatively explains the establishment of spiral magnetic ordering. Indeed, for o-$R$MnO$_3$ the ratio
\begin{equation}
\frac{J_{b}}{|J_{ab}|}>\frac{1}{2}
\label{eqSpiral}
\end{equation}
for $ab$ plane FM NN exchange $J_{ab}$ and AFM NNN exchange $J_b$ along the axis $b$ gives a spiral (with a propagation vector along the $b$ axis) as a magnetic ground state \footnote{1/2 appears in Eq.\ \ref{eqSpiral} instead of 1/4 for the case of an infinite spin chain as we take into account the number of equivalent bonds with exchange couplings $J_{b}$ and $J_{ab}$, which are equal to 2 and 4, respectively}. However, the source of E-AFM ordering is still under debate. For example, Kimura \textit{et al.} \cite{Kimura:2003kq} stated that the two-dimensional Heisenberg model with FM NN and certain competing AFM NNN couplings in the $ab$ plane can give E-AFM ordering, whereas Kaplan \cite{Kaplan:2009fx,Kaplan:pRvO8IfS} demonstrated that this state cannot occur in this model unless the biquadratic exchange interaction of the form
\begin{equation}
H_{bq}=\sum_{\langle k,l\rangle} j(\mathbf{S}_k\cdot \mathbf{S}_{l})^2
\label{eqBq}
\end{equation}
is included in the Hamiltonian.  In turn, Solovyev \cite{Solovyev:2009by} claimed that it is crucial to consider the exchange interaction between the third nearest neighbors in the $ab$ planes to stabilize the E-AFM state.

 \begin{figure}
\centering
{\includegraphics[scale=0.47, trim=1.5cm 0.3cm 0.5cm 0.3cm]{ABCetc1.eps}}
\caption
{(Color online) Collinear magnetic orderings (in Wollan-Koehler notation \cite{PhysRev.100.545}) of the Mn spins within the perovskite unit cell: (a) A-AFM, (b) G-AFM, (c) C-AFM and (d) FM. $J_c$ and $J_{ab}$ indicate the NN exchange couplings along the $c$ axis and within the $ab$ planes, respectively.}
\label{fig2}
\end{figure}

Aside from the disagreement on the source of the E-AFM order, the application of the Heisenberg model for a quantitative description of the magnetism in o-$R$MnO$_3$ gives ambiguous results. Assuming that the magnetism in o-$R$MnO$_3$ is fully described by the Heisenberg Hamiltonian and considering only the couplings between NN spins, the total energy can be written as
\begin{equation}
E=\sum_{\langle i,j\rangle}{J_{ij}\mathbf{S_i}\cdot \mathbf{S_j}+E_0,}
\end{equation}
where $E_0$ includes all other (nonmagnetic) interactions.
\begin{table}[b]
\caption{The exchange parameters $J_{ab}$ and $J_c$ (in meV) calculated with different theoretical approaches for LaMnO$_3$ using normalized values of spins $\mid \mathbf{S_{i}} \mid$=$\mid \mathbf{S_{j}} \mid$=1.}
\begin{tabular}{p{64pt}|p{40pt}p{40pt}|p{40pt}p{40pt}}
\hline
\hline
& \multicolumn{2}{p{84pt}|}{\centering{$J_{ab}$}} & \multicolumn{2}{p{84pt}}{\centering{$J_{c}$}} \\
\multirow{-2}{60pt}{\centering{Method}} &  \centering{$\Delta E_{AG}$} & \centering{$\Delta E_{FC}$} & \centering{$\Delta E_{FA}$} & \centering{$\Delta E_{CG}$} \tabularnewline
\hline
\centering{GGA-PBE \cite{Yamauchi:2008ia}} & \centering{-27.7} & \centering{-22.5} & \centering{0.5} & \centering{-10.0}  \tabularnewline
\centering{HFA \cite{Solovyev:2009by}} & \centering{-4.75} & \centering{-1.25} & \centering{10.0} & \centering{3.0}  \tabularnewline
\centering{GGA-PW91 \cite{Evarestov:2005ji}} & \centering{-18.0} & \centering{-14.6} & \centering{5.0} & \centering{-1.75}  \tabularnewline
\hline
\hline
\end{tabular}
\label{table1}
\end{table}
Therefore, one can see that the difference in the total energies of the unit cell of o-$R$MnO$_3$ with A-AFM and G-AFM orientations (see Fig.\ \ref{fig2}(a) and (b), respectively) of Mn$^{3+}$ spins ($\Delta E_{AG}$) defines the value of exchange coupling $J_{ab}$.  Moreover, this value should be the same as that given by the difference in the energies of FM and C-AFM (Fig.\ \ref{fig2}(d) and (c), respectively) states ($\Delta E_{FC}$). Similarly, the NN exchange $J_c$ along the $c$ axis can be extracted from the following energy differences: $\Delta E_{FA}=$ $E$(FM)-$E$(A-AFM) and $\Delta E_{CG}=$ $E$(C-AFM)-$E$(G-AFM) and the obtained values should be the same for these two cases. The energies of FM, A-AFM, C-AFM and G-AFM states have been calculated by several groups for the series of o-$R$MnO$_3$ applying different theoretical approaches \cite{{Yamauchi:2008ia}, {Solovyev:2009by}, {Evarestov:2005ji}}. Using the published values of these energies, we calculate for each case $\Delta E_{AG}$ and $\Delta E_{FC}$, which define $J_{ab}$ and should give the same results. However, we find that the obtained values $\Delta E$ are significantly different. The same is found for $J_c$, where $\Delta E_{FA}$ and $\Delta E_{CG}$ give in some cases even different signs. As an example, we present in Table \ref{table1} the values of $J_{ab}$ and $J_c$ in LaMnO$_3$ obtained using generalized gradient approximation in the form of Perdew, Burke and Ernzerhof (GGA-PBE)\cite{Yamauchi:2008ia}, Hartree-Fock approximation (HFA) \cite{Solovyev:2009by} and GGA with the Perdew-Wang-91 functional (GGA-PW91) \cite{Evarestov:2005ji}. We would like to point out, that we do not compare the values of $J_c$ and $J_{ab}$ obtained with different approximations and presented in different rows of the Table \ref{table1}. For each approximation we compare two values of $J_{ab}$ ($J_c$), which were obtained using $\Delta E_{AG}$ and $\Delta E_{FC}$ ($\Delta E_{FA}$ and $\Delta E_{CG}$) and, in principle, should give very similar values. Table \ref{table1} also demonstrates, that the inconsistencies in the values of exchanges are not related to the choice of the exchange-correlation potential. It should be noted, that the addition of the biquadratic term (Eq. \ref{eqBq}) in the Hamiltonian cannot explain these results as it cancels out in each energy difference.

These inconsistencies have not been addressed in the literature and require further investigation. On one hand, they could arise from the presence of strong NNN couplings. On the other hand, they could point to the presence of other significant  couplings beyond the Heisenberg Hamiltonian, which have to be taken into account for a proper theoretical analysis of the magnetism in o-$R$MnO$_3$.

\subsection{Mapping of DFT onto the Heisenberg model}
\label{sec2C}
In order to examine the relevance of the Heisenberg model for o-$R$MnO$_3$ and to clarify the inconsistencies in the previous theoretical results, we perform a thorough analysis of the microscopic exchange couplings by mapping the results of density functional theory (DFT) \cite{Hohenberg:1964ut,Kohn:1965ui} calculations onto the Heisenberg Hamiltonian. For that purpose we use two approaches, described below in this section and based on certain modifications of the initial magnetic states. In this context we point out, that in the DFT calculations periodic boundary conditions are applied and a variation of a state of a Mn spin on one site leads to the same variation of Mn spin states on all periodically equivalent sites.

\subsubsection{Calculations with collinear spin configurations}
\label{sec2c1}
The first approach, described in detail in Refs. \onlinecite{{Xiang:2011cn},{Whangbo:2005eg}}, is based on calculations of the total energy of the system with collinear spin alignment when the spin states on two sites (let us denote them as 1 and 2) within the given unit cell are modified. If the magnetism in the system is fully described by the Heisenberg Hamiltonian, the energy for such a spin pair can be written as follows:
\begin{equation}
E=nJ_{12}\mathbf S_1\cdot\mathbf S_2+\mathbf S_1\cdot\mathbf h_1+\mathbf S_2\cdot\mathbf h_2+E_{all}+E_0,
\label{eq_MappDFT}
\end{equation}
where $n$ is the number of equivalent bonds with exchange coupling $J_{12}$, which connect ions 1 and 2 (necessary to take into account the periodic boundary conditions), $\mathbf h_1=\sum_{i\ne 1,2}J_{1i}\mathbf S_i,$  $\mathbf h_2=\sum_{i\ne 1,2}J_{2i}\mathbf S_i$ and $E_{all}=\sum_{i,j\ne 1,2}J_{ij}\mathbf S_i\cdot \mathbf S_j$. The first term in Eq.\ \ref{eq_MappDFT} describes Heisenberg exchange interactions between spins in the considered pair, the second (third) term corresponds to the coupling of the spin 1 (2) with all other spins in the unit cell except spin 2 (1), $E_{all}$ characterizes the exchange couplings between all spins in the unit cell apart from spins 1 and 2, and $E_0$ contains other (nonmagnetic) energy contributions. It is important to use a reasonably large supercell to include in the analysis all couplings which could be significant in a considered system. Four different collinear configurations of the spins 1 and 2 are possible - up-up, up-down, down-up and down-down and their energies can be calculated using DFT. Then, the exchange interaction between these spins can be found using the formula:
\begin{equation}
J_{12}=\frac{E_{\uparrow\uparrow}+E_{\downarrow\downarrow}-E_{\uparrow\downarrow}-E_{\downarrow\uparrow}}{4nS^2}.
\label{j12}
\end{equation}
For a more direct comparison with other materials we prefer not to normalize our reported values of $J$ by $S^2$ (thus we set $S=1$ for Mn). Substituting in this expression the energies $E_{\uparrow\uparrow}$, $E_{\downarrow\downarrow}$, $E_{\uparrow\downarrow}$ and $E_{\downarrow\uparrow}$ using Eq.\ 8, one sees that all terms, except those describing the exchange interaction between spins 1 and 2, cancel out. As a result, the parameter $J_{12}$ should not depend on the orientation of the spins of the remaining ions in the unit cell.

\subsubsection{Noncollinear calculations}
\label{sec2c2}
This approach is based on the calculation of the total energy of the system when some spins are rotated away from an initial collinear state \cite{Novak:2008di} and can be illustrated by the example of the unit cell with 4 magnetic ions.
We consider A-AFM ordering for spins in the unit cell as a starting point and rotate the spins of ions 2 and 4 (see Fig.\ \ref{fig2}(a)) by an angle $\alpha$ keeping them antiparallel to each other until we reach G-AFM ordering (Fig.\ \ref{fig2} (b)). The energy of the system as a function of $\alpha$ within the Heisenberg model can be written as:
\begin{equation}
E(\alpha)=-4J_cS^2+8J_{ab}S^2\cos\alpha+E_0,
\end{equation}
and can be calculated using DFT. The resulting curve should fit the form
\begin{equation}
f(\alpha)=A_1+B_1\cos\alpha
\label{eqFit}
\end{equation}
if the Heisenberg model provides an accurate description (independently of the number of considered exchange couplings as periodic boundary conditions are applied) and the fitting parameter $B_1$ should define the exchange coupling constant $J_{ab}=B_1/8S^2$. $J_c$ can be extracted similarly by rotating spins on sites 3 and 4 from G-AFM to C-AFM ordering (from Fig.\ \ref{fig2} (b) to (c), respectively).

\subsection{Computational details}
We perform spin-polarized electronic structure calculations using the Vienna \textit{Ab initio} Simulation Package (VASP) \cite{Kresse:1996vf} within the projector-augmented plane wave (PAW) method of DFT. We use the GGA+U approximation for the exchange-correlation potential (in the form of Perdew, Burke and Ernzerhof \cite{Perdew:1996ug}) and apply the parameter of on-site Coulomb repulsion for Mn $d$ states of $U$=2 eV. The parameter of the effective on-site exchange (Hund's rule) interaction $J_H$ is always set to zero. We consider only isotropic exchange interactions, thus spin-orbit coupling is not included in our calculations. To eliminate the effects from the ordering of the f-electron moments of rare-earth ions, we use pseudopotentials for $R$ with the $f$ states frozen in the core. The value of the energy cutoff is set to 600 eV. In all calculations the experimental crystal structures \cite{Alonso:2000kv, Okamoto:2008ey, Wdowik:2011kz} are considered if it is not otherwise specified. The structures are kept fixed to isolate the contributions from spin-lattice coupling.
To construct the set of projected Wannier functions \cite{Ku:2002bf} we use the Wannier90 \cite{Mostofi:2008ff} code and the VASP2WANNIER90 interface \cite{Franchini:2012fl}.

\section{Deviation from Heisenberg model}
\label{sec3}
 \begin{figure}
\centering
{\includegraphics[scale=0.61, trim=1.5cm 0.3cm 0.5cm 0.3cm]{exchanges.eps}}
\caption
{(Color online) Heisenberg interactions in TbMnO$_3$, which are considered in this work. Mn atoms within the 40 atom supercell are highlighted with dark purple. Light  purple circles indicate Mn ions which belong to the neighboring supercells. NN exchanges are indicated in red, NNN in blue.}
\label{figEx}
\end{figure}
\begin{table}[b]
\caption{Calculated values of NN and NNN exchange interactions (in meV) in TbMnO$_3$ for FM and A-AFM cases.}
\begin{tabular}{p{40pt}p{30pt}p{30pt}p{30pt}p{30pt}p{30pt}p{30pt}}
\hline
\hline
 & \centering{$J_c$} &\centering{$J_{ab}$} & \centering{$J_a$} & \centering{$J_{diag}$} & \centering{$J_b$} & \centering{$J_3$} \tabularnewline
\hline
\centering{FM} & \centering{3.68} & \centering{-4.62} & \centering{-0.06} & \centering{0.97} & \centering{1.10} & \centering{1.11} \tabularnewline
\centering{A-AFM} & \centering{-0.85} & \centering{-5.16} & \centering{-0.32} & \centering{-0.10} & \centering{0.68} & \centering{1.26} \tabularnewline
\hline
\hline
\end{tabular}
\label{tab2}
\end{table}
\begin{table}[b]
\caption{The values of the exchange coupling constant $J_c$ (in meV) in TbMnO$_3$ calculated using the magnetic states shown in Fig.\ \ref{fig40at}. $E_{4sp}$ indicates the contributions to the values of $J_c$ from four-spin ring exchange $K$ between Mn spins confined in adjacent $ab$ planes.}
\begin{tabular}{p{45pt}p{45pt}p{45pt}p{45pt}p{45pt}}
\hline
\hline
 & \centering{(a)} &\centering{(b)} & \centering{(c)} & \centering{(d)} \tabularnewline
\hline
\centering{$J_c$} & \centering{$-0.88$} & \centering{$-0.68$} & \centering{$0.92$} & \centering{$2.84$}  \tabularnewline
\centering{$E_{4sp}$} & \centering{$-4K$} & \centering{$-4K$} & \centering{0} & \centering{$4K$}  \tabularnewline
\hline
\hline
\end{tabular}
\label{Jc}
\end{table}
We start with the analysis of the microscopic exchange couplings in the most studied multiferroic orthorhombic perovskite compound TbMnO$_3$ \cite{Xiang:2008jv,Malashevich:2008ew}. We initially assume that the magnetism in this material is fully described by the Heisenberg Hamiltonian (Eq.\ \ref{eqHeis}) and limit ourselves to consideration of the exchange couplings up to third NN within the $ab$ planes and second NN between $ab$ planes. In our notation $J_c$ and $J_{ab}$ are the NN exchanges along the $c$ axis and in the $ab$ plane (see Fig.\ \ref{figEx}) respectively; $J_a$ corresponds to the second NN exchange along the $a$ direction, $J_b$ - along the $b$ axis,  $J_{diag}$ couples second NN in adjacent $ab$ planes; $J_3$ is an exchange between third NN in the $ab$ planes. We extract these parameters applying the method described in Sec.\ \ref{sec2c1}. For this purpose we consider an 80 atom supercell (the orthorhombic unit cell \cite{Alonso:2000kv} is doubled in the $a$ and $b$ directions) and a $\Gamma$-centered  $3\times3\times 5$ k-point mesh. For each $J$ we choose the corresponding spin pair in the supercell and calculate the total energies of the system for the four possible orientations of spins in this pair (up-up, up-down, down-up, down-down). We keep the rest of Mn spins fixed first in the FM state (FM case) and then in the A-AFM state (A-AFM case). The calculated values of $J$ are presented in Table \ref{tab2}.

We find, that $J_b$ is rather weak relative to $J_{ab}$ for both (FM and A-AFM) cases and according to Eq.\ \ref{eqSpiral} cannot produce the spiral state in TbMnO$_3$. The other possible source of frustration could be the AFM coupling $J_3$ which is stronger even than the second NN in-plane couplings $J_a$ and $J_b$. The importance of $J_3$ was already pointed out in Ref.\  \onlinecite{Solovyev:2009by}. Note, that Ref.\ \onlinecite{Solovyev:2009by} proposed strong or weak $J_3$ couplings depending on the relative orientation of the occupied $d$ orbitals on the interacting sites. However, we obtain the same value of $J_3$ for both possible orbital orientations.

\begin{figure}
\centering
{\includegraphics[scale=0.41]{whangbo4str.eps}}
\caption
{(Color online) 40 atom supercell of TbMnO$_3$ (side view) with magnetic orders which were used to calculate the exchange parameter $J_c$. Tb and O ions are not shown.}
\label{fig40at}
\end{figure}
The key result of these calculations is that the values of the exchanges, especially $J_c$, have different magnitudes and in some cases even different signs for FM and A-AFM cases while within the Heisenberg description they should be equal (or at least very similar).
To double check this result and to determine its origin, we extract $J_c$, which shows the largest inconsistency, with the same method using a 40 atom supercell (20 atom unit cell doubled along the $b$ axis, $7\times 4\times 5$ $\Gamma$-centered k-point mesh). We calculate the total energies switching the direction of spins 1 and 2, but now the rest of spins are kept in the states shown in Fig.\ \ref{fig40at}. The new calculated $J_c$ values are presented in Table \ref{Jc}. One can see that the $J_c$ value obtained using state (a) (which is A-AFM order) is in agreement with the $J_c$ value which was found using the 80 atom supercell and starting from the same magnetic state. Interestingly, the values of $J_c$ are similar for the states (a) and (b), where the closest surroundings of spins 1 and 2 are identical. In turn, if the states differ by the direction of one spin in the nearest neighborhood of the considered spin pair (such as between states (a) and (c) or (c) and (d)), $J_c$ changes by approximately the same amount (in average by 1.85 meV). This suggests the presence of strong couplings beyond the Heisenberg Hamiltonian which involve in some way the magnetic interactions between the nearest-neighboring Mn spins.
\begin{figure}
\centering
{\includegraphics[trim=1cm 0cm 0cm 0cm,scale=0.52]{AtoG_tbmnu2.eps}}
\caption
{(Color online) Dependence of the energy $E$ (relative to the energy of A-AFM order) of TbMnO$_3$ on the rotation angle $\alpha$ of spins from A-AFM to G-AFM state. The results of DFT calculations are shown by dots and the fitting to the Heisenberg model (Eq.\ \ref{eqFit}) by the blue line. The red line indicates the fitting to a Hamiltonian which includes bilinear and higher order exchange couplings.}
\label{fitToHeis}
\end{figure}

\begin{figure}
\includegraphics[scale=0.42,trim=1cm 0cm 0cm 0cm]{analysis.eps}
\caption{(Color online) (a) Ideal cubic perovskite structure; (b) Purely JT distorted structure (tetragonal); (c) Fully JT+GFO distorted structure (orthorhombic) of TbMnO$_3$.}
\label{figAnalysis}
\end{figure}
Another check can be done by the method described in Sec.\ \ref{sec2c2}. To apply this method we consider a 20 atom unit cell and a $5\times 5\times 3$ k-point mesh. We rotate the spins on sites 2 and 4 from A-AFM to G-AFM ordering (see Fig.\ \ref{fig2}) and calculate the energies $E$ of the system (relative to the energy of A-AFM order) for several values of spin rotation angle $\alpha$ between 0$^{\circ}$ and 180$^{\circ}$. $E(\alpha)$ and its fitting to $f(\alpha)$ (Eq.\ \ref{eqFit}) are presented in Fig.\ \ref{fitToHeis} (black dots and blue line, respectively). One can see that $E(\alpha)$ shows clear deviation from the cosinusoidal behaviour predicted by the Heisenberg model.

These results lead us to the conclusion that the Heisenberg Hamiltonian in its usual form is not able to accurately describe the magnetism in TbMnO$_3$ and more couplings have to be taken into consideration.

\section{Origin of non-Heisenberg behavior}
\label{sec4}
\subsection{Orbital ordering}
First, we investigate whether the observed non-Heisenberg behavior can originate from the presence of the orbital ordering in TbMnO$_3$. Indeed, as was already described in Sec.\ \ref{sec:2b}, the magnetic and orbital orderings are related as far as the coupling of spins on neighboring Mn$^{3+}$ ions is determined by the occupation of their particular orbitals through the superexchange mechanism. To take this behavior into account, the ordinary superexchange was generalized for the case of systems with orbital degeneracy by Kugel and Khomskii \cite{Kugel:1973ub}. They introduced a model Hamiltonian, in which, besides the Heisenberg exchange, they included terms describing orbital-orbital and orbital-spin couplings. The latter gives the change in the orbital ordering with variation of the spin alignment (or vice versa) and, if it is large enough, can explain the different values of exchange obtained for A-AFM and FM cases as well as the deviation from Heisenberg behavior observed in noncollinear calculations.

The occupied $e_g$ orbital $|\psi \rangle$ for each Mn$^{3+}$ ion is uniquely defined in terms of the orbital mixing angle $\theta$ by Eq.\ \ref{orbmixing}.
To extract $\theta$ for TbMnO$_3$, so as to trace its evolution with structural distortions and to estimate the strength of coupling between orbitals and spins we perform the following analysis: We construct a perfect cubic perovskite structure for TbMnO$_3$ using a 20 atom unit cell and keeping the volume of each MnO$_6$-octahedron equal to the experimental one (see Fig.\ \ref{figAnalysis} (a)). Then we start to apply the JT modes Q$_2$ and Q$_3$ (without GFO distortion) in such a way that $Q_{i,applied}=aQ_{i,exp}$ (thus the angles $\varphi$, which are defined by Eq.\ \ref{eq5}, are equal for all values of $a$). $a$ is varied from 0 to 1 and $Q_{i,exp}$ corresponds to the structure with the full JT distortion (Fig.\ \ref{figAnalysis} (b)). To reach a more transparent description of the orbital ordering, we use a representation in terms of Wannier functions \cite{Marzari:2012eu} (WF), which, unlike Bloch functions, are localized in space and have minimal overlap with the surrounding orbitals. To construct a set of $e_g$ like WFs, we proceed similarly to Refs.\ \onlinecite{Yin:2006hd, Ku:2002bf}. We calculate the Bloch functions within GGA+U (using the structures corresponding to different $a$), and for each structure we define an energy window, in which $e_g$ bands are located, based on projected densities of states and band structures. Then, using the VASP2WANNIER90 interface and Wannier90 code, we construct four WFs via projection of atomiclike $|d_{z^2}\rangle$ and $|d_{x^2-y^2}\rangle$ orbitals centered on two Mn sites (1 and 2) on the majority spin Bloch bands within the chosen energy window.  Then we calculate the occupation matrices in the basis of these WFs for several values of $a$ and two types of ordering of the Mn magnetic moments (A- and G-AFM). Solving the eigenvalue problem for these matrices, we find $|\psi_i \rangle$ ($i=1,2$) and, thus, $\theta_i$. We plot $\theta_1$ as a function of $a$ starting from $a$=0.4 (see Fig.\ \ref{mixangles}, left half of the graph), since smaller amplitudes of JT distortion give a metallic ground state. $\theta_2$ has the same values as $\theta_1$, but the opposite sign.

\begin{figure}[t]
\centering
{\includegraphics[scale=0.5]{OMA_both.eps}}
\caption
{(Color online) Orbital mixing angle as a function of the amplitudes of JT and GdFeO$_3$-type distortions for G-AFM and A-AFM magnetic orderings in TbMnO$_3$. In the part of the graph highlighted with violet (cyan), only the amplitude of JT (GFO)-distortion is varied. $\varphi$ is determined using Eq.\ \ref{eq5}.}
\label{mixangles}
\end{figure}

\begin{figure}
\centering
{\includegraphics[scale=0.36, trim=0.6cm 0cm 0cm 0cm]{GG1.eps}}
\caption
{(Color online) Dependence of the energy $E$ (relative to the energy of A-AFM order) on the rotation angle $\alpha$ of spins from A-AFM to G-AFM state for the structures of TbMnO$_3$ with different amplitudes of JT and GFO-distortions. Dots indicate the results of DFT calculations, lines show the fittings to the Heisenberg Hamiltonian (Eq.\ \ref{eqFit}). (a) and (b) correspond to the structures with 40\% and 100\% JT distortion, respectively, without octahedral tiltings; (c) and (d) to the structures with the full JT distortion and 60\% and 100\% GFO distortion, respectively. Plot (d) was obtained using the crystal structure which unlike the experimental one does not include the antiferroelectric displacements of $R$ cations, thus it is not identical to the one shown on Fig.\ \ref{fitToHeis}.}
\label{figJTGFO}
\end{figure}

As we expected, the calculated $\theta$ are different for A- and G-AFM orderings. With increasing JT distortion from 40 to 100\% the mixing angles for both AFM orders change by approximately $6^\circ$ and tend to the value of $\varphi\approx 114^\circ$ which is imposed by the structure ($\varphi$ is calculated using Eq.\ \ref{eq5} and experimental lengths of long, medium and short bonds in Mn-O octahedra). It is important, that the difference in the orbital mixing angles  $\Delta \theta$ between A- and G-AFM orderings (in other words, the variation of the orbital ordering by the change in the magnetic structure) is quite small for the whole range of JT distortion amplitudes and reaches a maximum of $\Delta\theta\approx 3^\circ$. In order to check, whether such a small variation of the orbital mixing angle $\Delta\theta$ can cause the deviation from Heisenberg behavior
which was found in Sec.\ \ref{sec3}, we perform the calculations of the total energies rotating the spins from A-AFM to G-AFM ordering and using the structures with different amplitudes of JT distortion ($a=0.4$ and 1). The obtained angular dependences of the total energy and their fittings to $f(\alpha)$ (Eq.\ \ref{eqFit}) are shown in Fig.\ \ref{figJTGFO} (a), (b). It is clearly seen that the calculated $E(\alpha)$ fit well with the Heisenberg Hamiltonian for both amplitudes of JT distortion. Therefore, one can conclude, that the variation in $\theta$ associated with the change in the magnetic order is not sufficient to explain the large deviation from the Heisenberg model which was observed in our previous calculations. It should be taken into account, however, that the energy scale is 4-6 times larger than in the case where we perform the calculations using the experimental crystal structure (Fig.\ \ref{fitToHeis}). This is because in the latter case the exchange energy is reduced by the presence of GFO distortion. Thus, it is also possible, that the contribution from $\Delta\theta$ is not significant in comparison with the strong exchange energy within the tetragonal structure, but could be important when the orthorhombic distortion comes into play. Therefore, we are motivated to analyse next the effect of GFO distortion on the orbital and magnetic orderings.

To investigate the variation of the orbital mixing angle by GFO distortion we again construct four projected WFs. In this case, to initialize projections, we introduce a local coordinate system for each MnO$_6$ octahedron in such a way that $x$, $y$ and $z$ axes are aligned as much as possible along the long, short and medium Mn-O bonds, respectively. Other than that, we proceed in the same way as before: Starting from the fully JT distorted structure (Fig.\ \ref{figAnalysis} (b)),  we gradually increase the octahedral rotations to reach the experimentally observed Mn-O-Mn bond angles. The final structure is shown in Fig.\ \ref{figAnalysis} (c); in comparison with the experimental structure, this one does not include a small antiferroelectric shift of Tb cations. We calculate the orbital mixing angles as a function of the amplitude of GFO distortion for A- and G-AFM orderings (Fig.\ \ref{mixangles}, right part of the graph). Then we perform spin rotations from A- to G-AFM ordering with 60\% and 100\% GFO distorted crystal structures. Corresponding angular dependences of the total energy are presented in Fig.\ \ref{figJTGFO} (c), (d).

We find, that increasing GFO distortion causes smaller variation of the orbital mixing angle ($\approx 1.5-3^\circ$) in comparison with JT distortion for both magnetic orderings. Moreover, it almost does not affect $\Delta \theta$ between different types of magnetic ordering (indeed, curves for A-AFM and G-AFM stay almost parallel). However, it induces and enhances the deviation of $E(\alpha)$ from $f(\alpha)$ as shown in Fig.\ \ref{figJTGFO} (c), (d). Therefore we conclude that non-Heisenberg behavior originates from the modification of Mn-O-Mn bond angles due to the reduction of the energy of the exchange interactions between NN Mn spins, which makes weak energy contributions more significant.

\begin{figure}
\centering
{\includegraphics[scale=0.345, trim=1cm 1cm 2cm 1cm]{withoutOO1.eps}}
\caption
{(Color online) Dependence of the energy $E$ (relative to the energy of A-AFM order) on the rotation angle $\alpha$ of spins from A-AFM to G-AFM state for compounds without orbital ordering: (a) TbCrO$_3$ and (b) TbFeO$_3$. Dots correspond to the results of DFT calculations, lines show the fittings to the Heisenberg Hamiltonian (Eq.\ \ref{eqFit}).}
\label{figCrFe}
\end{figure}

In order to check whether these weak contributions are provided by $\Delta \theta$, we perform the same spin rotations for two compounds which do not contain JT active ions (therefore, do not have an orbital ordering), TbCrO$_3$ (Cr$^{3+}$: $t^3_{2g}e^0_g$) and TbFeO$_3$ (Fe$^{3+}$: $t^3_{2g}e^2_g$). In these calculations we use the structure of TbMnO$_3$ with Mn$^{3+}$ replaced by Cr$^{3+}$ and Fe$^{3+}$, respectively, which allows us to modify the strength of NN exchange interactions by changing the occupation of $d$ orbitals without any variation of the crystal structure. Indeed, in TbCrO$_3$ magnetic couplings are mostly provided by the hopping processes between $t_{2g}$ orbitals (mediated by oxygen $p$ states) as $e_g$ orbitals are empty, whereas in TbFeO$_3$ both $t_{2g}$ and $e_g$ states of one spin direction are fully occupied and participate in exchange interactions. Moreover, the $e_g$ orbitals in octahedral coordination have stronger overlap with O $p$ states than the $t_{2g}$ due to their geometry, and therefore provide stronger coupling. As a result, one can expect significantly larger magnitudes of NN exchanges for TbFeO$_3$ than for TbCrO$_3$.

The calculated $E(\alpha)$ are presented in Fig.\ \ref{figCrFe}. Both compounds exhibit deviations of $E(\alpha)$ from the cosinusoidal behavior even in the absence of an orbital ordering. Notably, the deviation is stronger for the case of TbCrO$_3$ than for TbFeO$_3$. We assume that in TbCrO$_3$ the couplings which are not considered in the Heisenberg Hamiltonian are comparable in magnitude with NN exchanges between Mn $t_{2g}$ states and cause a stronger deviation from cosinusoidal behavior, whereas in TbFeO$_3$ they are not significant relative to the strong NN exchange and the Heisenberg model works sufficiently well. Thus, the observed non-Heisenberg behavior cannot be explained by the presence of the orbital ordering. Instead, it appears in the materials where the energy of the exchange couplings is reduced by the modification of Mn-O-Mn bond angles or by the occupation of the orbitals participating in the superexchange.

\subsection{Higher order exchange couplings}
\label{sec4b}
Next we investigate whether exchange couplings of higher order than the usual bilinear term might be responsible for the observed deviation from the Heisenberg model. Generally speaking, the higher order exchanges as well as the bilinear coupling can be derived from a half-filled Hubbard model in the limit $t/U<<1$ (which is applicable for insulators),
\begin{equation}
\label{eq19}
H=-t\sum\limits_{\langle i,j\rangle,\sigma}\left(c_{i\sigma}^{\dagger}c_{j\sigma}+c_{j\sigma}^{\dagger}c_{i\sigma}\right)+U\sum\limits_{j}\hat{n}_{j\uparrow}\hat{n}_{j\downarrow},
\end{equation}
where $t$ is a hopping parameter, $U$ describes the on-site Coulomb repulsion, $c_{j\sigma}^{\dagger}$ and $c_{j\sigma}$ are operators of creation and annihilation of electrons with spin $\sigma$ in the Wannier state $w(\mathbf{r}-\mathbf{R}_j)$ and $\hat{n}_{j\uparrow}=c^\dagger_{j\uparrow}c_{j\uparrow}$ is the occupation number operator.  Second order perturbation theory in $t$ gives the energy correction in the form of Heisenberg exchange, whereas the fourth order gives biquadratic, four-spin ring interactions and additional contributions to NNN couplings. The four-spin ring term describes the consecutive hopping processes between NN ions forming a four-site plaquette and has the following form \cite{{Fazekasbook},{Takahashi:1976vn}}:
\begin{eqnarray}
H_{4sp} & \propto &  \left[ \left(\mathbf S_i\cdot\mathbf S_j\right)\left(\mathbf S_k\cdot\mathbf S_l\right)+  \left(\mathbf S_i\cdot\mathbf S_l\right)\left(\mathbf S_k\cdot\mathbf S_j\right) \right.  \nonumber \\
&-&  \left. \left(\mathbf S_i\cdot\mathbf S_k\right)\left(\mathbf S_j\cdot\mathbf S_l\right)\right],
\label{4body}
\end{eqnarray}
where $i$, $j$, $k$ and $l$ enumerate spins of the plaquette.
For the Heisenberg model to be valid, all higher order terms should be negligible compared with the bilinear term.  As their strength is defined by $t^4/U^3$, and that of the bilinear term by $t^2/U$, this should be the case in the limit of small enough $t/U$. However, several theoretical and experimental groups found that in some compounds these terms are significant. For example, it was shown, that the results of a paramagnetic resonance study \cite{Harris:2011wd} of pairs of Mn$^{2+}$ ions in MgO fit much better with a Hamiltonian that includes biquadratic exchange $H_{bq}$ (Eq.\ \ref{eqBq}) than with the ordinary Heisenberg Hamiltonian. Later, the significance of $H_{bq}$ was invoked to explain the establishment of the E-AFM ordering \cite{Kaplan:2009fx} in o-$R$MnO$_3$ with $R$=Ho...Lu as we mentioned in Sec.\ \ref{sec:2b}.  The four-spin ring interaction was found to be important to explain the dispersion of the magnetic excitations in La$_2$CuO$_4$ measured using inelastic neutron scattering experiments \cite{Coldea:2001gu}. It was also shown to be significant in the spin-ladder cuprates SrCu$_2$O$_3$, CaCu$_2$O$_3$ and Sr$_2$CuO$_4$ using \textit{ab initio} quantum chemistry embedded cluster calculations \cite{Calzado:2003iy}.

First, let us check whether the presence of the higher order terms can explain the inconsistent values of the exchange coupling constant $J_c$ in TbMnO$_3$ which were obtained in Sec.\ \ref{sec3} starting from different states with collinear spin alignment.
We already mentioned, that the addition of $H_{bq}$ cannot affect the resulting values of exchanges as the applied method considers the energy differences between states with collinear spin orientations and in these differences biquadratic terms always cancel out. Fourth order contributions to NNN interactions, if present, are already included in the analysis as they cannot be distinguished from the bilinear NNN couplings. To introduce the terms describing the four-spin ring exchanges we have to consider the couplings between spins in the plaquettes confined in the $ab$ planes as well as from those that contain pairs of Mn spins from neighboring $ab$ planes. We denote the corresponding coupling constants as $G$ and $K$ (see Fig.\ \ref{bqCalc} (a)). Thus, we can write the energies $E_{\uparrow \uparrow}$, $E_{\uparrow \downarrow}$, $E_{\downarrow \uparrow}$ and $E_{\downarrow \downarrow}$ for the 80 atom supercell of TbMnO$_3$ in A-AFM and FM cases including four-spin interactions and put them in Eq.\ \ref{j12} to extract $J_c$. In this way we find that the in-plane ring exchanges $G$ cancel each other for both cases in the linear combinations of these energies and obtain (for $S=1$)
\begin{equation}
\label{jcwith4b}
\begin{gathered}
J_c\mathrm{(A-AFM)} \to J_c-4K \\
J_c\mathrm{(FM)} \to J_c+4K.
\end{gathered}
\end{equation}
This result shows that the presence of $K$ is the most likely origin of the difference in the obtained values of $J_c$. Using Eqs.\ \ref{jcwith4b} and values of $J_c(\mathrm{A-AFM})$ and $J_c(\mathrm{FM})$ which were obtained in Sec.\ \ref{sec3}, one finds the value of $K\approx 0.6$ meV for TbMnO$_3$.
\begin{figure}
\centering
{\includegraphics[scale=0.4,trim= 0.8cm 0cm 0cm 0cm]{leastsqr.eps}}
\caption
{(Color online) Energies of the 80 atom supercell of TbMnO$_3$ with 54 inequivalent collinear magnetic configurations (referred to the energy of the A-AFM state) predicted by (a) the pure Heisenberg Hamiltonian and (b) the Hamiltonian, which includes bilinear and four-spin ring couplings, and plotted versus the energies of corresponding states calculated using DFT. Ideally, the model and DFT energies should be equal and points should lie on the dashed line. Insets show the deviations of the model energies from those calculated using DFT. Each bar corresponds to one considered magnetic configuration.}
\label{leastsquares}
\end{figure}
\begin{table*}[!htb]
\caption{Coupling constants (in meV) for bilinear, four-spin ring and biquadratic exchanges in TbMnO$_3$ (for $U$=2 eV and $U$=1 eV), PrMnO$_3$ (for $U$=1eV) and LuMnO$_3$ (for $U$=1 eV).}
\begin{tabular}{p{52pt}p{32pt}p{38pt}p{38pt}p{38pt}p{38pt}p{38pt}p{38pt}p{38pt}p{38pt}p{38pt}p{38pt}}
\hline
\hline
 & \centering{$U_\mathrm{Mn}$} & \centering{$J_c$} & \centering{$J_{ab}$} & \centering{$J_a$} &\centering{$J_{diag}$} & \centering{$J_b$} & \centering{$J_3$} & \centering{$G$} & \centering{$K$} & \centering{$j_c$} & \centering{$j_{ab}$}
\tabularnewline
\hline
\centering{TbMnO$_3$} & \centering{2 eV} & \centering{1.22} & \centering{-6.01} & \centering{-0.47} & \centering{0.31} & \centering{0.65} &\centering{1.21} & \centering{-0.05} & \centering{0.50} & \centering{-0.81} & \centering{-2.29}
\tabularnewline
\hline
\centering{PrMnO$_3$} & \centering{1 eV} & \centering{1.79} & \centering{-14.16} & \centering{-0.48} & \centering{0.88} & \centering{0.26} &\centering{3.25} & \centering{-0.07} & \centering{0.80} & \centering{-2.61} & \centering{-2.95}
\tabularnewline
\centering{TbMnO$_3$} & \centering{1 eV} & \centering{4.26} & \centering{-3.86} & \centering{-0.37} & \centering{0.58} & \centering{0.85} &\centering{1.77} & \centering{-0.02} & \centering{0.77} & \centering{-0.47} & \centering{-2.09}
\tabularnewline
\centering{LuMnO$_3$} & \centering{1 eV} & \centering{3.76} & \centering{-0.48} & \centering{-0.55} & \centering{0.53} & \centering{0.93} &\centering{1.75} & \centering{0.15} & \centering{0.66} & \centering{-0.37} & \centering{-2.29}
\tabularnewline
\hline
\hline
\end{tabular}
\label{allcouplings}
\end{table*}
The same can be done for the 40 atom supercell of TbMnO$_3$ with magnetic configurations shown in Fig.\ \ref{fig40at}. As before, here we find that the in-plane four-spin couplings $G$ cancel each other. The contributions to $J_c$ arising from the inter-plane ring exchanges obtained for these states are summarized in Table \ref{Jc}. One can see the relation between these contributions and the values of $J_c$ which were calculated with DFT using the structures (a)-(d) and presented in the first line of Table \ref{Jc}.
Indeed, for the states (a) and (b), the interplane four-spin exchanges contribute exactly the same to $J_c$ ($E_{4sp}=-4K$), and the values of $J_c$ which we extracted using DFT for these states are very similar. States (a) and (c) as well as (c) and (d)  have contributions to $J_c$ which differ by $4K$. Notably, the $J_c$ values which we extracted for these states vary by approximately the same amount (in average $1.85$ meV). This gives the value of $K\approx 0.5$ meV, which is in agreement with the value of $K$ obtained using Eqs.\ \ref{jcwith4b}. Thus we confirm the presence of the strong four-spin interplane exchange couplings in TbMnO$_3$ and show that the addition of these couplings to the model Hamiltonian can explain the inconsistent values of NN exchanges which were found in Sec.\ \ref{sec3}.

The size of the in-plane four-spin coupling can be estimated similarly by choosing the appropriate collinear spin states and calculating energy differences for them. However, we proceed in a different way. As we already calculated the total energies of the 80 atom supercell of TbMnO$_3$ for a large number (namely 54) of inequivalent magnetic collinear states, we can write the energies of these states using the model Hamiltonian that includes bilinear and four-spin ring couplings and construct an overdetermined system of linear equations, where the unknowns are the exchange coupling constants (bilinear ones (see Fig.\ \ref{figEx}): $J_c$, $J_{ab}$, $J_a$, $J_{diag}$, $J_b$, $J_3$ and four-spin ones (Fig.\ \ref{bqCalc} (a)): $G$ and $K$). To build this system of equations we use only the states which are insulating and take the energy of the A-AFM state as a reference. Then we use the least mean square method to extract all coupling constants. The obtained values are presented in Table \ref{allcouplings}. We find that the in-plane four-spin coupling $G$ is negligible in comparison with the inter-plane one $K$. Further investigation is required to find an explanation for this observation.

Using our extracted values of the coupling constants, we calculate the expected energies of all 54 states using the considered model Hamiltonian. We plot them versus the energies of these states (referred to the energy of the A-AFM state) calculated using DFT in order to examine how well our model predicts the magnetic properties of the system (ideally, model and DFT energies should be the same). The result is presented in Fig.\ \ref{leastsquares} (b). Similarly, we extract the coupling constants and calculate the energies of the magnetic states using the pure Heisenberg Hamiltonian (in an overdetermined system of equations, the only unknowns are the bilinear coupling constants:  $J_c$, $J_{ab}$, $J_a$, $J_{diag}$, $J_b$ and $J_3$). The model energies plotted versus the energies obtained from first-principles calculations are shown in Fig.\ \ref{leastsquares} (a). Moreover, we extract the deviations of the energies predicted by both Hamiltonians from their values obtained with DFT for each considered magnetic state. These deviations are summarized in the bar charts shown in the insets in Fig.\ \ref{leastsquares}. One can see that the Hamiltonian which includes both bilinear and four-spin terms gives much better agreement with the results of DFT calculations than the pure Heisenberg Hamiltonian. We repeat this analysis also for the Hamiltonian which involves six bilinear exchange couplings and only inter-plane four-spin ring coupling $K$, as $G$ was found to be negligible. The extracted coupling constants as well as the deviations between model and DFT energies remain almost the same as those which were obtained using the full Hamiltonian (which includes also $G$). This means that the addition of just one parameter $K$ into the model Hamiltonian can already significantly improve the description of the magnetism in the considered compound.

The effect of the higher order exchange interactions in TbMnO$_3$ can also be examined using noncollinear calculations. The simplest approach is to map the angular dependence of the total energy, obtained in Sec.\ \ref{sec3} from the spin rotations from A-AFM to G-AFM orderings, to the Hamiltonian which includes bilinear, biquadratic and four-spin ring interactions by fitting to the function
\begin{equation}
g(\alpha)=A_2+B_2\cos(\alpha)+C_2\cos^2(\alpha).
\end{equation}
The result is shown in Fig.\ \ref{fitToHeis} and clearly demonstrates that the introduction of the higher order couplings into the model Hamiltonian greatly improves the fitting. The strengths of these couplings are determined by the fitting parameter $C_2$. Note that this term includes the contributions from four-spin ring exchanges as well as from the in-plane biquadratic couplings and that these terms cannot be separated. Similar behavior of $E(\alpha)$ was found in Ref. \onlinecite{Novak:2008di} for hexagonal YMnO$_3$ using \textit{ab initio} calculations within the LDA+U approximation, where it was discussed only in terms of bilinear and biquadratic exchanges.
\begin{figure}
\centering
{\includegraphics[scale=0.5,trim=1cm 0cm 0cm 0cm]{bqex2.eps}}
\caption
{(Color online) Magnetic orderings which are used to extract the biquadratic exchange interactions: (a) $j_{ab}$ in the $ab$ planes ($G$ and $K$ indicate four-spin ring exchange couplings in plaquettes of Mn spins confined in the $ab$ planes and those containing pairs of spins from neighboring $ab$ planes, respectively); (b) $j_c$ along the $c$ axis (violet dashed rectangle indicates a rotation plane of the spin 4).}
\label{bqCalc}
\end{figure}
\label{sec5}
\begin{figure*}[!htb]
\begin{minipage}{0.329\linewidth}
\includegraphics[width=1\linewidth,trim=1cm 0cm 0.5cm 0cm]{leastsqrPrMn.eps} \\
\end{minipage}
\hfill
\begin{minipage}{0.328\linewidth}
\includegraphics[width=1\linewidth,trim=1cm 0cm 0.5cm 0cm]{TbMnleastsqr.eps} \\
\end{minipage}
\hfill
\begin{minipage}{0.329\linewidth}
\includegraphics[width=1\linewidth,trim=1cm 0cm 0.5cm 0cm]{LuMnleastsqr.eps} \\
\end{minipage}
\caption{(Color online) Energies of the 80 atom supercells of PrMnO$_3$ (a,b), TbMnO$_3$ (c,d) and LuMnO$_3$ (e,f) with more than 30 inequivalent magnetic configurations (referred to that of the lowest-energy state) predicted by the pure Heisenberg Hamiltonian (blue dots) and the Hamiltonian which includes bilinear and four-spin ring exchanges (red dots) and plotted versus the energies of corresponding states calculated using DFT.}
\label{allrmn}
\end{figure*}

To complete the analysis of the full model Hamiltonian we need to estimate the coupling constants which define the biquadratic exchanges in TbMnO$_3$. For this we only take into account the biquadratic interactions between the nearest neighbors in $ab$ planes and along the $c$ axis and denote the corresponding coupling constants as $j_{ab}$ and $j_c$ (see Fig.\ \ref{bqCalc}). The problem can be simplified if we eliminate the contribution from four-spin inter-plane ring exchanges by considering the magnetic states which set to zero at least one scalar product in each of the three terms in Eq.\ \ref{4body}. This can be achieved by setting three Mn spins in the unit cell perpendicular to each other as shown in Fig.\ \ref{bqCalc}. By rotation of the remaining spin the angular dependence of the total energy can be obtained and the coupling constants can be found from the corresponding fittings.
To extract $j_{ab}$ we start from the magnetic state shown in Fig.\ \ref{bqCalc} (a) and rotate spin 4 by an angle $\alpha$ from 0 to $180^\circ$ in the $ac$-plane. The energy of this system can be written as follows:
\begin{eqnarray}
E(\alpha) & = & E+4J_{ab} \cos(\alpha)+8J_{diag} \sin(\alpha)+ \nonumber \\  &+& 4j_{ab}\cos^2(\alpha)+8G\cos^2(\alpha),
\end{eqnarray}
where the third term is given by spins 1 and 4 and all other terms by spins 3 and 4 ($S$=1) and $E$ includes the exchange couplings which are constant at every $\alpha$ for the considered magnetic states and other nonmagnetic interactions. The coupling constants can be extracted by fitting to the function:
\begin{eqnarray}
f(\alpha)&=&A_1+4J_{ab}\cos(\alpha)+8J_{diag}\sin(\alpha)+ \nonumber \\ &+&(D_1+8G)\cos^2(\alpha),
\end{eqnarray}
where we set the values of $J_{ab}$, $J_{diag}$ and $G$ to those which were extracted in the collinear calculations and presented in the first line of Table \ref{allcouplings}. $D_1/4$ defines $j_{ab}=-2.29$ meV.
We proceed in a similar way to extract the coupling constant $j_c$. Starting from the magnetic state presented in Fig.\ \ref{bqCalc} (b) and rotating spin 4 by $\alpha$ from 0 to 180$^\circ$ in the $ac$ plane, we obtain $E(\alpha)$. One can see, that in the considered magnetic state neither in-plane nor interplane four-spin ring couplings contribute to $E(\alpha)$ and fitting to
\begin{eqnarray}
g(\alpha)& =& A_2+2J_c \cos(\alpha)+8J_{diag} \sin(\alpha)+\nonumber \\ &+&D_2\cos^2(\alpha),
\end{eqnarray}
gives $j_c=D_2/2=-0.81$ meV.

Thus we demonstrate that the higher order exchange interactions are significant in TbMnO$_3$ (especially the four-spin ring interplane coupling $K$ and biquadratic in-plane coupling $j_{ab}$) and have to be included in the model Hamiltonian to properly describe the magnetic properties of this material.

\section{Exchange interactions in other o-$R$MnO$_3$}
\label{sec5}
Finally, in this section we investigate the evolution of the exchange couplings in o-$R$MnO$_3$ with increasing GFO distortion due to decrease in the radius of the $R$ cation. For this purpose we consider PrMnO$_3$ and LuMnO$_3$, which have among the largest and the smallest $R$ radii in the series of o-$R$MnO$_3$, respectively. We set $U_\mathrm{Mn}=1$ eV for Mn $d$ states, which gives a correct magnetic ground state for both considered systems. We calculate the total energies of these systems within the 80 atom supercells (experimental unit cells \cite{{Wdowik:2011kz},{Okamoto:2008ey}} are duplicated along $a$ and $b$ directions) for 34 and 32 inequivalent collinear magnetic states, respectively. By writing the expressions for the energies of these magnetic states using the model Hamiltonian that includes bilinear and four-spin ring exchange interactions, we obtain overdetermined systems of equations with respect to six bilinear and two four-spin ring couplings for each compound. In both cases the lowest energy state was taken as the reference (the A-AFM state for PrMnO$_3$ and the E-AFM state for LuMnO$_3$). We solve these systems of equations using the least mean square method and find the values of all coupling constants (Table \ref{allcouplings}). For comparison, we calculate the exchange couplings in a similar way for TbMnO$_3$ for the considered value of $U_\mathrm{Mn}$.
From Table \ref{allcouplings} one can see that the increasing GFO distortion has the strongest effect on the in-plane NN coupling $J_{ab}$ which changes by more than one order of magnitude from Pr to Lu. It also enhances $J_b$ as we expected (see Sec.\ \ref{sec:2b}). However, the behavior of the other coupling constants (particularly the absence of a trend in the variation of $J_c$ and $J_a$, and the strong change in some coupling constants and weak change in others) with the variation of Mn-O-Mn bond angles still requires a further analysis.

As the next step, we use the extracted values of the exchange couplings to calculate the energies of all considered magnetic states within this model Hamiltonian and plot them versus the energies of these states which we calculate using DFT (see Fig.\ \ref{allrmn} (b), (d) and (f)). For comparison, we extract similarly the coupling constants using the pure Heisenberg Hamiltonian for each compound, then calculate the energies of all states, predicted by this Hamiltonian, and plot them versus the DFT energies of these states (Fig.\ \ref{allrmn} (a), (c) and (e))). The insets in all resulting graphs show the deviations of the model energy from the DFT energy for each considered state.

From Fig.\ \ref{allrmn} we find that the model Hamiltonian which includes both the bilinear and four-spin ring exchanges gives much better agreement with the results of the first-principles calculations. One can conclude that the Heisenberg model works relatively well for PrMnO$_3$, since the model and DFT energies almost coincide in Fig.\ \ref{allrmn} (a). However, if one compares the $\Delta E$ values which were obtained for the bilinear-only case (see insets in Fig.\ \ref{allrmn} (a), (c) and (e)), one can see that they are similar and even larger than those of TbMnO$_3$ and LuMnO$_3$, but small relative to the  energy scale of the bilinear couplings, in particular, $J_{ab}$ (see Table \ref{allcouplings}). When GFO distortion increases and $J_{ab}$ drops (as in TbMnO$_3$ and LuMnO$_3$), the $\Delta E$ due to the non-Heisenberg terms become significant. However, when the four-spin ring couplings are added in the model Hamiltonian (Fig.\ \ref{allrmn} (b), (d), (f)), the $\Delta E$ values reduce drastically.

Finally, we extract the biquadratic couplings $j_c$ and $j_{ab}$ for PrMnO$_3$, TbMnO$_3$ and LuMnO$_3$ (see Table \ref{allcouplings}) applying the method which was described in detail at the end of Sec.\ \ref{sec4b}. For all compounds we obtain strong negative in-plane biquadratic couplings $j_{ab}$, which favor collinear alignment of spins within the $ab$ planes and can drive an evolution of a magnetic order from a spiral to an E-AFM state for systems with large GFO distortions. This confirms the finding of Ref.\ \onlinecite{Kaplan:2009fx}, where the biquadratic exchange interaction was claimed to be important in the establishment of the E-AFM order. The in-plane coupling $j_c$ is found to be much more affected by GFO distortion than $j_{ab}$. Again, the origin of this behavior still has to be clarified.

Thus we show that the Heisenberg Hamiltonian cannot accurately predict the magnetic properties of o-$R$MnO$_3$ with large GFO distortions. In these materials the bilinear couplings  become comparable in magnitude with the biquadratic and four-spin ring interactions and it is essential to include the latter two into the model Hamiltonian for proper analysis of the magnetism.

\section{Summary}
\label{sec6}
In summary, we investigated the microscopic exchange couplings in the series of o-$R$MnO$_3$ in order to find an isotropic part of a model Hamiltonian which can properly describe the magnetism in these materials. The work was motivated by the inconsistencies in the results obtained in several theoretical studies when the exchange couplings in o-$R$MnO$_3$ was mapped onto the Heisenberg Hamiltonian as well as by the absence of agreement on the origin of the E-AFM order in o-$R$MnO$_3$ with small $R$ cations ($R$=Ho...Lu). We started our analysis from the most studied multiferroic orthorhombic manganite, TbMnO$_3$, and estimated the exchange couplings with several approaches (collinear and noncollinear) using DFT. We observed a clear deviation from the behavior predicted by the Heisenberg model. Moreover, we confirmed the importance of the AFM third NN in-plane coupling, $J_3$, in the establishment of the spiral state in this compound.
In the next step we explored whether the observed non-Heisenberg behavior originates from the presence of the orbital ordering in TbMnO$_3$ and its coupling with the Mn spins. To check this, we analyzed the changes in the orbital mixing angle with structural distortions (Jahn-Teller and GFO) and with variation of the magnetic ordering using the Wannier function representation. We found that the orbital mixing angle indeed can be affected by the magnetic order, however, we showed that this change is quite small and is almost unchanged by the structural distortions. In turn, we found that the deviation from Heisenberg behavior does not appear when the amplitude of JT distortion is varied. It appears only with increasing GFO distortion, which decreases the energy of the NN exchange interactions and makes the weak energy contributions more important. We demonstrated, however, that these weak contributions do not originate from the variation of the orbital ordering. Indeed, compounds which do not have an orbital degree of freedom (such as TbCrO$_3$ and TbFeO$_3$) also exhibit a deviation from the energy behavior predicted by the Heisenberg model. Finally, we investigated the effects of exchange couplings of higher order than the ordinary bilinear exchange (biquadratic and four-spin ring interactions), which are usually neglected. We demonstrated that the higher order contributions are significant (especially inter-plane four-spin ring exchange $K$ and biquadratic in-plane coupling $j_{ab}$) and can be comparable with the bilinear exchanges for o-$R$MnO$_3$ with small radii of $R$ cations. We showed that the inconsistent values of the exchange couplings which were obtained from the collinear calculations within the Heisenberg model (Sec.\ \ref{sec3}) can be explained only by addition of the four-spin ring couplings into the model Hamiltonian. Moreover, we proved that such a model Hamiltonian predicts the magnetic properties of o-$R$MnO$_3$ with much higher precision than the pure Heisenberg Hamiltonian. The finding of the strong negative in-plane biquadratic exchange interaction $j_{ab}$, which favors a collinear spin alignment within the $ab$ planes, is in agreement with the suggestion of Ref.\ \onlinecite{Kaplan:2009fx} that $H_{bq}$ is crucial in the establishment of the E-AFM state in o-$R$MnO$_3$ with small radii of the $R$ cations.

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